Mistakes Notebook

SAT Diagnostic SAT Information and Ideas 2026-07-30 skipped

Which choice best describes the narrator's view of his expedition to the North Pole?

Answer: Absurd but necessary

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SAT Diagnostic SAT Statistics 2026-07-30 skipped

Nick surveyed a random sample of the freshman class of his high school to determine whether the Fall Festival should be held in October or November. Of the $90$ students surveyed, $25.6%$ preferred October. Based on this information, about how many students in the entire $225$-person class would be expected to prefer having the Fall Festival in October?

Answer: $60$

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SAT Diagnostic SAT Algebra 2026-07-30 missed ×2 skipped

$f(x) = 7x + 1$ The function gives the total number of people on a company retreat with $x$ managers. What is the total number of people on a company retreat with 7 managers?

Answer: 50

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SAT Diagnostic SAT Algebra 2026-07-30 skipped

The system of equations $\dfrac{1}{2}y = 4x - 1$ and $\dfrac{1}{2}y = 2$ has solution $(x, y)$. What is the value of $x$ ?

Answer: $6$

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SAT Diagnostic SAT Algebra 2026-07-30 skipped

In the equation $\dfrac{a}{b} = \dfrac{a}{c}$ above, if $a$ is negative and $b$ is positive, which of the following must be true?

Answer: $c > 1$

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SAT Diagnostic SAT Ratios, Rates and Percentages 2026-07-30 skipped

One method of calculating the approximate age, in years, of a tree of a particular species is to multiply the diameter of the tree, in inches, by a constant called the growth factor for that species. The table gives the growth factors for eight species of trees: Red maple $4.5$, River birch $3.5$, Cottonwood $2.0$, Black walnut $4.5$, White birch $5.0$, American elm $4.0$, Pin oak $3.0$, Shagbark hickory $7.5$. According to the information in the table, what is the approximate age of an American elm tree with a diameter of $12$ inches?

Answer: $48$ years

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SAT Diagnostic SAT Craft and Structure 2026-07-30 skipped

As used in line 4, “state” most nearly refers to a

Answer: political entity.

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SAT Diagnostic SAT Information and Ideas 2026-07-30 missed ×2 skipped

"The Young Girl" is a 1920 short story by Katherine Mansfield. In the story, the narrator takes an unnamed seventeen-year-old girl and her younger brother out for a meal. In describing the teenager, Mansfield frequently contrasts the character's pleasant appearance with her unpleasant attitude, as when Mansfield writes of the teenager, ______ Which quotation from "The Young Girl" most effectively illustrates the claim?

Answer: "While we waited she took out a little, gold powder-box with a mirror in the lid, shook the poor little puff as though she loathed it, and dabbed her lovely nose."

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SAT Diagnostic SAT Craft and Structure 2026-07-30 skipped

Which choice best describes how Burke would most likely have reacted to Paine’s remarks in the final paragraph of Passage 2?

Answer: With disapproval, because changing conditions are insufficient justification for changing the form of government.

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SAT Diagnostic SAT Advanced Math 2026-07-30 missed ×2 skipped

$\dfrac{14x}{7y} = 2 \sqrt{w + 19}$ The given equation relates the distinct positive real numbers $w$, $x$, and $y$. Which equation correctly expresses $w$ in terms of $x$ and $y$?

Answer: $w = \left(\dfrac{x}{y}\right)^2 - 19$

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SAT Diagnostic SAT Advanced Math 2026-07-30 missed ×2 skipped

$f(x) = 5470(0.64)^{\dfrac{x}{12}}$ The function $f$ gives the value, in dollars, of a certain piece of equipment after $x$ months of use. If the value of the equipment decreases each year by $p$% of its value the preceding year, what is the value of $p$?

Answer: 36

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SAT Diagnostic SAT Geometry 2026-07-30 missed ×2 skipped

Two identical rectangular prisms each have a height of 90 centimeters (cm). The base of each prism is a square, and the surface area of each prism is $K$ cm${}^2$. If the prisms are glued together along a square base, the resulting prism has a surface area of $\dfrac{92}{47} K$ cm${}^2$. What is the side length, in cm, of each square base?

Answer: 8

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SAT Diagnostic SAT Information and Ideas 2026-07-30 skipped

Which choice best supports the claim that Quilotoa was not responsible for the Little Ice Age?

Answer: Lines 71-75 ("But . . . closer match")

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SAT Diagnostic SAT Expression of Ideas 2026-07-30 missed ×2 skipped

While researching a topic, a student has taken the following notes: - The *Atlantic Monthly* magazine was first published in 1857. - The magazine focused on politics, art, and literature. - In 2019, historian Cathryn Halverson published the book *Faraway Women and the "Atlantic Monthly."* - Its subject is female authors whose autobiographies appeared in the magazine in the early 1900s. - One of the authors discussed is Juanita Harrison. The student wants to introduce Cathryn Halverson's book to an audience already familiar with the *Atlantic Monthly*. Which choice most effectively uses relevant information from the notes to accomplish this goal?

Answer: Cathryn Halverson's *Faraway Women and the "Atlantic Monthly"* discusses female authors whose autobiographies appeared in the magazine in the early 1900s.

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SAT Diagnostic SAT Information and Ideas 2026-07-30 skipped

According to figure 1, in 2017, the cost of which of the following fuels is projected to be closest to the 2009 US average electricity cost shown in figure 2?

Answer: Advanced nuclear

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TMUA 2022 Paper 1 TMUA Trigonometric Equations 2026-07-29 missed ×2 skipped

How many real solutions are there to the equation $2 \cos^4 \theta - 5 \cos^2 \theta + 3 = 0$ in the interval $0 \leq \theta \leq 2 \pi$ ?

Answer: 3

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TMUA 2022 Paper 1 TMUA Coordinate Geometry 2026-07-29 missed ×2 skipped

Find the complete set of values of $p$ for which the equation $x^2 - 2 p x + y^2 - 6 y - p^2 + 8 p + 9 = 0$ describes a circle in the $x y$-plane.

Answer: $p < 0$ or $p > 4$

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TMUA 2022 Paper 1 TMUA Integration 2026-07-29 missed ×2 skipped

Given the following statements about a function f - $f''(x) = a$ for all $x$ - $f(0) = 1$, $f(1) = 2$ find the value of $a$.

Answer: $6$

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TMUA 2022 Paper 1 TMUA Plane Geometry 2026-07-29 missed ×2 skipped

These sectors of circles are similar. The arc length of the smaller sector is 6. The difference between the areas of the sectors is 21. Find the positive difference between the perimeters of the sectors.

Answer: 8

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TMUA 2022 Paper 1 TMUA Sequences and Series 2026-07-29 missed ×2 skipped

The terms $x_n$ of a sequence follow the rule $x_{n+1} = \dfrac{x_n + p}{x_n + q}$ where $p$ and $q$ are real numbers. Given that $x_1 = 3$, $x_2 = 5$, and $x_3 = 7$, find the value of $x_4$

Answer: $13$

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TMUA 2022 Paper 1 TMUA Integration 2026-07-29 missed ×2 skipped

Given that $\displaystyle\int_{\log_2 5}^{\log_2 20} x d x = \log_2 M$ what is the value of $M$?

Answer: 100

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TMUA 2022 Paper 1 TMUA Integration 2026-07-29 missed ×2 skipped

Find the finite area enclosed between the line $y = 0$ and the curve $y = x^2 - 4 |x| - 12$

Answer: $144$

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TMUA 2022 Paper 1 TMUA Sequences and Series 2026-07-29 missed ×2 skipped

A geometric sequence has first term $a$ and common ratio $r$, where $a$ and $r$ are positive integers and $r$ is greater than 1. The sum of the first $n$ terms of this sequence is denoted by $S_n$ It is given that the terms of the sequence satisfy $S_30 - S_20 = k S_10$ for some positive integer $k$. What is the smallest possible value of $k$ ?

Answer: $2^{20}$

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TMUA 2022 Paper 1 TMUA Algebraic Manipulations 2026-07-29 missed ×2 skipped

This question is about pairs of functions f and g that satisfy $f(x) - g(x) = 2 \sin x$ $f(x) g(x) = \cos^2 x$ for all real numbers $x$. Across all solutions for $f(x)$, what is the minimum value that $f(x)$ attains for any $x$?

Answer: $-2$

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TMUA 2022 Paper 1 TMUA Functions and Their Graphs 2026-07-29 missed ×2 skipped

A sequence of translations is applied to the graph of $y = x^3$ Which of the following graphs could be the result of this sequence of translations? I $\ y = x^3 - 3 x^2 + 9 x - 27$ II $\ y = x^3 - 9 x^2 + 27 x - 3$ III $\ y = 27 x^3 - 9 x^2 + x - 3$

Answer: II only

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TMUA 2022 Paper 1 TMUA Exponentials and Logarithms 2026-07-29 missed ×2 skipped

Evaluate $\displaystyle\sum_{n=1}^{100} \log_10 (3^{1-n})$

Answer: $-4950 \log_10 3$

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TMUA 2022 Paper 1 TMUA Functions and Their Graphs 2026-07-29 missed ×2 skipped

A family of quadratic curves is given by $y_k = 2 \left(x - \dfrac{k}{2}\right)^2 + \dfrac{k^2}{2} + 4 k + 3$ where $k$ is any real number and $y_k$ is a function of $x$. All these curves are sketched, and the point with the lowest $y$-coordinate among all the curves $y_k$ is $(a, b)$. Find the value of $a + b$

Answer: $-7$

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TMUA 2022 Paper 1 TMUA Algebraic Manipulations 2026-07-29 missed ×2 skipped

Given that $\left(a^3 + \dfrac{2}{b^3}\right) \left(\dfrac{2}{a^3} - b^3\right) = \sqrt{2}$ where $a$ and $b$ are real numbers, what is the least value of $a b$?

Answer: $-\sqrt{2}$

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TMUA 2022 Paper 1 TMUA Plane Geometry 2026-07-29 missed ×2 skipped

A circle has centre $O$ and radius 6. $P$, $Q$ and $R$ are points on the circumference with angle $\text{POQ} = \dfrac{2 \pi}{3}$ The area of the triangle $\text{POQ}$ is $9 \sqrt{3}$ What is the greatest possible area of triangle $\text{PRQ}$?

Answer: $27 \sqrt{3}$

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TMUA 2022 Paper 1 TMUA Differentiation 2026-07-29 missed ×2 skipped

A rectangle is drawn in the region enclosed by the curves $p$ and $q$, where $p(x) = 8 - 2 x^2$ $q(x) = x^2 - 2$ such that the sides of the rectangle are parallel to the $x$- and $y$-axes. What is the maximum possible area of the rectangle?

Answer: $\dfrac{40 \sqrt{10}}{9}$

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TMUA 2022 Paper 1 TMUA Equations 2026-07-29 missed ×2 skipped

The solutions to $7 x^4 - 6 x^2 + 1 = 0$ are $\pm \cos \theta$ and $\pm \cos \beta$. Which one of the following equations has solutions $\pm \sin \theta$ and $\pm \sin \beta$ ?

Answer: $7 x^4 - 8 x^2 + 2 = 0$

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TMUA 2022 Paper 1 TMUA Plane Geometry 2026-07-29 missed ×2 skipped

Find the complete set of values of $x$ for which there are two non-congruent triangles with the side lengths and angle as shown in the diagram.

Answer: $3 < x < 4$

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TMUA 2022 Paper 1 TMUA Curve Sketching 2026-07-29 missed ×2 skipped

It is given that $f(x) = x^2 (x-1)^2 (x-2)$ $g(x) = -p (x-q)^2 (x-r)^2$ where $p$, $q$ and $r$ are positive and $q < r$ Find the set of values of $q$ and $r$ that guarantees the greatest number of distinct real solutions of the equation $f(x) = g(x)$ for all $p$.

Answer: $q < 1$ and $1 < r < 2$

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The equation $f(x) = g(x)$ is a polynomial equation with degree 5, so has at most 5 distinct real solutions; we wish there to be 5 such solutions for all values of $p$. We would have fewer than 5 distinct real solutions if we either have a square factor of the form $(x - a)^2$ for some $a$, or if the graph of $y = f(x) - g(x)$ has fewer than four turning (stationary) points, or if this graph has a minimum point above the $x$-axis or a maximum point below it, for then it would not cross the $x$-axis 5 times. Let us write $ h(x) = f(x) - g(x) = x^2(x-1)^2(x-2) + p(x-q)^2(x-r)^2 $ We can try expanding and differentiating this, but that looks horrible. Let us think about the above description, though, remembering that there need to be 5 distinct real solutions for all values of $p$. The graph of $y = f(x) = x^2(x-1)^2(x-2)$ intersects the $x$-axis at $x = 0$ (where it is tangent to the axis, because it is a double root), $x = 1$ (likewise) and $x = 2$, and it is positive when $x > 2$ and negative or zero when $x < 2$, so the graph looks like this: The graph of $y = g(x) = -p(x-q)^2(x-r)^2$ has two points of intersection with the $x$-axis, one at $x = q$ and one at $x = r$, and at both of these points, the graph is tangent to the axis. (In the case $q = r$, there is just one point of tangency.) The graph is entirely below or touching the $x$-axis as we are given that $p > 0$. It looks like this: If $r > 2$, then the graph of $y = g(x)$ will entirely miss one of the "valleys" of $y = f(x)$ if $p$ is large, as it will if $r < 1$; in this case, there would be at most three points of intersection for large $p$. So we must have $1 < r < 2$. If $1 < q < r$, then it will miss the left valley for large enough $p$, so we must have $0 < q < 1$ and $1 < r < 2$. In this case, the graph of $y = g(x)$ must intersect both valleys twice as $g(1) < 0$: This sketch shows four points of intersection, and we are looking for five. But the graph of $y = f(x)$ tends to $-oo$ as $x$ tends to $-oo$ far faster that $y = g(x)$, regardless of the value of $p$, as $y = f(x)$ is a quintic but $y = g(x)$ is only a quartic. So there will always be another point of intersection with $x < 0$. Hence the correct answer is option B. Note that there is actually a small simplification in the above argument. We showed that we cannot have $1 < q < r$ and deduced that we must have $0 < q < 1$. But what about the possibility $q = 1$? In that case, there would be a double root at $x = 1$, and so there would not be five distinct roots, hence we cannot have $q = 1$. But what about $r$? We showed that we cannot have $r > 2$ or $r < 1$. If $r = 1$, we have the same problem, but what about $r = 2$ and $0 < q < 1$? In this case, it is still true that the equation has five distinct real solutions for all positive $p$, as $x = 2$ is only a single root of $f(x) = 0$. So the full set is actually $q < 1$ and $1 < r \leq 2$.

TMUA 2022 Paper 1 TMUA Counting and Probabilities 2026-07-29 missed ×2 skipped

Circle $C_1$ is defined as $x^2 + y^2 = 25$ A second circle $C_2$ has radius 4 and centre $(a, b)$ where $-2 \leq a \leq 2$ and $-3 \leq b \leq 3$ If the centre of $C_2$ is equally likely to be located anywhere within the given range, what is the probability that $C_2$ intersects $C_1$ ?

Answer: $\dfrac{24 - \pi}{24}$

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TMUA 2022 Paper 1 TMUA Curve Sketching 2026-07-29 missed ×2 skipped

$n$ is the number of points of intersection of the graphs $y = |x^2 - a^2|$ and $y = a^2 |x - 1|$ where $a$ is a real number. What is the smallest value of $n$ that is **not** possible?

Answer: $n = 2$

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The graph $y = |x^2 - a^2|$ intersects the $x$-axis at $\pm a$. The graph $y = a^2|x - 1|$ intersects the $x$-axis at $x = 1$. Replacing $a$ with $-a$ does not change the equations, so we may as well assume that $a \geq 0$. Here is a sketch of the graphs with $a = 2$; we note that the graphs always intersect at $x = 0$ (and $y = a^2$ there): We immediately see that when $a > 1$, there will be 4 points of intersection: the graph $y = a^2|x-1|$ is below the "hill" at $(1, 0)$, so intersects it twice as $x$ moves away from $x = 1$, and then intersects the quadratic $y = x^2 - a^2$ for some $x < -a$ and some $x > a$ as the quadratic grows faster than the linear $y = -a^2(x - 1)$ or $y = a^2(x - 1)$ in these regions. What happens when $a = 1$? In this case, the equations become $y = |x^2 - 1|$ and $y = x - 1$. The derivative of the quadratic $y = x^2 - 1$ is $2x$, so the gradient is just over 2 when $x$ is just over 1. (There is no gradient at the point $x = 1$.) Therefore the graph looks like this: We see that we have lost one point of intersection: there are now only three points of intersection. What happens if $0 < a < 1$? Now the vertex of $y = a^2|x-1|$ will be to the right of $x = a$, where the quadratic meets the axis, and so there will be no points of intersection there -- the quadratic grows too quickly to meet the linear function for $x > 1$. (To be more precise: the gradient of the quadratic to the right of $x = a$ is $2x > 2a > a^2$, so the quadratic will be growing faster than the linear function.) On the left of $x = a$, we have one point of intersection to the left of $x = -a$ as before. There is also a point of intersection at $(0, a^2)$. Since this is the vertex of the quadratic, the linear function must go below the "hill", giving another two points of intersection, yielding four points of intersection in total: There is one remaining case to consider: $a = 0$. In this case, the graphs become $y = |x^2| = x^2$ and $y = 0$, and these graphs have a single point of intersection at the origin. Therefore the possible numbers of points of intersection are 1, 3 and 4, and the smallest number not possible is $n = 2$, option B.

Stewart Precalc 6e Chapter 3 Review Precalculus Quadratic Function - Standard Form 2026-07-29 missed ×2 skipped

A quadratic function is given. (a) Express the function in standard form. (b) Graph the function. $f(x) = x^2 + 4 x + 1$.

Answer:

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Stewart Precalc 6e Chapter 3 Review Precalculus Quadratic Function - Standard Form 2026-07-29 missed ×2 skipped

A quadratic function is given. (a) Express the function in standard form. (b) Graph the function. $f(x) = -2 x^2 + 12 x + 12$.

Answer:

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Stewart Precalc 6e Chapter 3 Review Precalculus Quadratic Function - Standard Form 2026-07-29 missed ×2 skipped

A quadratic function is given. (a) Express the function in standard form. (b) Graph the function. $g(x) = 1 + 8 x - x^2$.

Answer:

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Stewart Precalc 6e Chapter 3 Review Precalculus Quadratic Function - Standard Form 2026-07-29 missed ×2 skipped

A quadratic function is given. (a) Express the function in standard form. (b) Graph the function. $g(x) = 6 x - 3 x^2$.

Answer:

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Stewart Precalc 6e Chapter 3 Review Precalculus Quadratic Function - Maximum/Minimum 2026-07-29 missed ×2 skipped

Find the maximum or minimum value of the quadratic function $f(x) = 2 x^2 + 4 x - 5$.

Answer:

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Stewart Precalc 6e Chapter 3 Review Precalculus Quadratic Function - Maximum/Minimum 2026-07-29 missed ×2 skipped

Find the maximum or minimum value of the quadratic function $g(x) = 1 - x - x^2$.

Answer:

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Stewart Precalc 6e Chapter 3 Review Precalculus Quadratic Application - Projectile Motion 2026-07-29 missed ×2 skipped

A stone is thrown upward from the top of a building. Its height (in feet) above the ground after $t$ seconds is given by the function $h(t) = -16 t^2 + 48 t + 32$. What maximum height does the stone reach?

Answer:

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Stewart Precalc 6e Chapter 3 Review Precalculus Quadratic Application - Profit Maximization 2026-07-29 missed ×2 skipped

The profit $P$ (in dollars) generated by selling $x$ units of a certain commodity is given by the function $P(x) = -1500 + 12 x - 0.004 x^2$. What is the maximum profit, and how many units must be sold to generate it?

Answer:

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Stewart Precalc 6e Chapter 3 Review Precalculus Polynomial Graph Transformation 2026-07-29 missed ×2 skipped

Graph the polynomial by transforming an appropriate graph of the form $y = x^n$. Show clearly all $x$- and $y$-intercepts. $P(x) = -x^3 + 64$.

Answer:

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Stewart Precalc 6e Chapter 3 Review Precalculus Polynomial Graph Transformation 2026-07-29 missed ×2 skipped

Graph the polynomial by transforming an appropriate graph of the form $y = x^n$. Show clearly all $x$- and $y$-intercepts. $P(x) = 2 x^3 - 16$.

Answer:

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Stewart Precalc 6e Chapter 3 Review Precalculus Polynomial Graph Transformation 2026-07-29 missed ×2 skipped

Graph the polynomial by transforming an appropriate graph of the form $y = x^n$. Show clearly all $x$- and $y$-intercepts. $P(x) = 2 (x + 1)^4 - 32$.

Answer:

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Stewart Precalc 6e Chapter 3 Review Precalculus Polynomial Graph Transformation 2026-07-29 missed ×2 skipped

Graph the polynomial by transforming an appropriate graph of the form $y = x^n$. Show clearly all $x$- and $y$-intercepts. $P(x) = 81 - (x - 3)^4$.

Answer:

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Stewart Precalc 6e Chapter 3 Review Precalculus Polynomial Graph Transformation 2026-07-29 missed ×2 skipped

Graph the polynomial by transforming an appropriate graph of the form $y = x^n$. Show clearly all $x$- and $y$-intercepts. $P(x) = 32 + (x - 1)^5$.

Answer:

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Stewart Precalc 6e Chapter 3 Review Precalculus Polynomial Graph Transformation 2026-07-29 missed ×2 skipped

Graph the polynomial by transforming an appropriate graph of the form $y = x^n$. Show clearly all $x$- and $y$-intercepts. $P(x) = -3 (x + 2)^5 + 96$.

Answer:

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Stewart Precalc 6e Chapter 3 Review Precalculus Remainder Theorem 2026-07-29 missed ×2 skipped

Find the indicated value of the polynomial using the Remainder Theorem. $Q(x) = x^4 + 4 x^3 + 7 x^2 + 10 x + 15$; find $Q(-3)$

Answer: $Q(-3) = 21$.

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Stewart Precalc 6e Chapter 3 Review Precalculus Factor Theorem 2026-07-29 missed ×2 skipped

Show that $\dfrac{1}{2}$ is a zero of the polynomial $P(x) = 2 x^4 + x^3 - 5 x^2 + 10 x - 4$

Answer: $P\left(\dfrac{1}{2}\right) = 0$.

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Stewart Precalc 6e Chapter 3 Review Precalculus Factor Theorem 2026-07-29 missed ×2 skipped

Use the Factor Theorem to show that $x + 4$ is a factor of the polynomial $P(x) = x^5 + 4 x^4 - 7 x^3 - 23 x^2 + 23 x + 12$

Answer: $P(-4) = 0$, so $x + 4$ is a factor.

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Stewart Precalc 6e Chapter 3 Review Precalculus Remainder Theorem 2026-07-29 missed ×2 skipped

What is the remainder when the polynomial $P(x) = x^{500} + 6 x^{201} - x^2 - 2 x + 4$ is divided by $x - 1$?

Answer: Remainder $= 8$.

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Stewart Precalc 6e Chapter 3 Review Precalculus Remainder Theorem 2026-07-29 missed ×2 skipped

What is the remainder when $x^{101} - x^4 + 2$ is divided by $x + 1$?

Answer: Remainder $= 0$.

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Stewart Precalc 6e Chapter 3 Review Precalculus Rational Zeros and Descartes' Rule 2026-07-29 missed ×2 skipped

A polynomial $P$ is given. (a) List all possible rational zeros (without testing to see whether they actually are zeros). (b) Determine the possible number of positive and negative real zeros using Descartes' Rule of Signs. $P(x) = x^5 - 6 x^3 - x^2 + 2 x + 18$

Answer: (a) Possible rational zeros: $\pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18$. (b) $0$ or $2$ positive real zeros; $1$ or $3$ negative real zeros.

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Stewart Precalc 6e Chapter 3 Review Precalculus Rational Zeros and Descartes' Rule 2026-07-29 missed ×2 skipped

A polynomial $P$ is given. (a) List all possible rational zeros (without testing to see whether they actually are zeros). (b) Determine the possible number of positive and negative real zeros using Descartes' Rule of Signs. $P(x) = 6 x^4 + 3 x^3 + x^2 + 3 x + 4$

Answer: (a) Possible rational zeros: $\pm 1, \pm 2, \pm 4, \pm \dfrac{1}{2}, \pm \dfrac{1}{3}, \pm \dfrac{2}{3}, \pm \dfrac{4}{3}, \pm \dfrac{1}{6}$. (b) $0$ positive real zeros; $0$, $2$, or $4$ negative real zeros.

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Stewart Precalc 6e Chapter 3 Review Precalculus Real Zeros and Graph 2026-07-29 missed ×2 skipped

A polynomial $P$ is given. (a) Find all real zeros of $P$, and state their multiplicities. (b) Sketch the graph of $P$. $P(x) = x^3 - 16 x$

Answer: Zeros: $0$, $4$, $-4$ (each multiplicity 1). $P(x) = x (x - 4)(x + 4)$.

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Stewart Precalc 6e Chapter 3 Review Precalculus Real Zeros and Graph 2026-07-29 missed ×2 skipped

A polynomial $P$ is given. (a) Find all real zeros of $P$, and state their multiplicities. (b) Sketch the graph of $P$. $P(x) = x^3 - 3 x^2 - 4 x$

Answer: Zeros: $0$, $4$, $-1$ (each multiplicity 1). $P(x) = x (x - 4)(x + 1)$.

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Stewart Precalc 6e Chapter 3 Review Precalculus Real Zeros and Graph 2026-07-29 missed ×2 skipped

A polynomial $P$ is given. (a) Find all real zeros of $P$, and state their multiplicities. (b) Sketch the graph of $P$. $P(x) = x^4 + x^3 - 2 x^2$

Answer: Zeros: $0$ (multiplicity 2), $-2$ (multiplicity 1), $1$ (multiplicity 1).

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Stewart Precalc 6e Chapter 3 Review Precalculus Real Zeros and Graph 2026-07-29 missed ×2 skipped

A polynomial $P$ is given. (a) Find all real zeros of $P$, and state their multiplicities. (b) Sketch the graph of $P$. $P(x) = x^4 - 5 x^2 + 4$

Answer: Zeros: $\pm 1$, $\pm 2$ (each multiplicity 1).

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Stewart Precalc 6e Chapter 3 Review Precalculus Real Zeros and Graph 2026-07-29 missed ×2 skipped

A polynomial $P$ is given. (a) Find all real zeros of $P$, and state their multiplicities. (b) Sketch the graph of $P$. $P(x) = x^4 - 2 x^3 - 7 x^2 + 8 x + 12$

Answer: Zeros: $-1$, $-2$, $2$, $3$ (each multiplicity 1). $P(x) = (x + 1)(x + 2)(x - 2)(x - 3)$.

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Stewart Precalc 6e Chapter 3 Review Precalculus Real Zeros and Graph 2026-07-29 missed ×2 skipped

A polynomial $P$ is given. (a) Find all real zeros of $P$, and state their multiplicities. (b) Sketch the graph of $P$. $P(x) = x^4 - 2 x^3 - 2 x^2 + 8 x - 8$

Answer: Real zeros: $2$, $-2$ (each multiplicity 1). Complex zeros: $1 \pm i$. $P(x) = (x - 2)(x + 2)(x^2 - 2 x + 2)$.

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Stewart Precalc 6e Chapter 3 Review Precalculus Real Zeros and Graph 2026-07-29 missed ×2 skipped

A polynomial $P$ is given. (a) Find all real zeros of $P$, and state their multiplicities. (b) Sketch the graph of $P$. $P(x) = 2 x^4 + x^3 + 2 x^2 - 3 x - 2$

Answer: Real zeros: $1$ and $-\dfrac{1}{2}$ (each multiplicity 1). Remaining factor: $2 x^2 + 2 x + 4$ (complex roots).

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Stewart Precalc 6e Chapter 3 Review Precalculus Real Zeros and Graph 2026-07-29 missed ×2 skipped

A polynomial $P$ is given. (a) Find all real zeros of $P$, and state their multiplicities. (b) Sketch the graph of $P$. $P(x) = 9 x^5 - 21 x^4 + 10 x^3 + 6 x^2 - 3 x - 1$

Answer: Real zeros: $1$ (multiplicity 3), $-\dfrac{1}{3}$ (multiplicity 2). $P(x) = 9 (x - 1)^3 \left(x + \dfrac{1}{3}\right)^2$.

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Stewart Precalc 6e Chapter 3 Review Precalculus Complex Numbers 2026-07-29 missed ×2 skipped

Evaluate the expression and write in the form $a + b i$. $(2 - 3 i) + (1 + 4 i)$

Answer: $3 + i$.

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Stewart Precalc 6e Chapter 3 Review Precalculus Complex Numbers 2026-07-29 missed ×2 skipped

Evaluate the expression and write in the form $a + b i$. $(3 - 6 i) - (6 - 4 i)$

Answer: $-3 - 2 i$.

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Stewart Precalc 6e Chapter 3 Review Precalculus Complex Numbers 2026-07-29 missed ×2 skipped

Evaluate the expression and write in the form $a + b i$. $(2 + i)(3 - 2 i)$

Answer: $8 - i$.

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Stewart Precalc 6e Chapter 3 Review Precalculus Complex Numbers 2026-07-29 missed ×2 skipped

Evaluate the expression and write in the form $a + b i$. $4 i \left(2 - \dfrac{1}{2} i\right)$

Answer: $2 + 8 i$.

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Stewart Precalc 6e Chapter 3 Review Precalculus Complex Numbers 2026-07-29 missed ×2 skipped

Evaluate the expression and write in the form $a + b i$. $\dfrac{4 + 2 i}{2 - i}$

Answer: $\dfrac{6}{5} + \dfrac{8}{5} i$.

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Stewart Precalc 6e Chapter 3 Review Precalculus Complex Numbers 2026-07-29 missed ×2 skipped

Evaluate the expression and write in the form $a + b i$. $\dfrac{8 + 3 i}{4 + 3 i}$

Answer: $\dfrac{41}{25} - \dfrac{12}{25} i$.

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Stewart Precalc 6e Chapter 3 Review Precalculus Complex Numbers 2026-07-29 missed ×2 skipped

Evaluate the expression and write in the form $a + b i$. $(1 + i)^3$

Answer: $-2 + 2 i$.

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Stewart Precalc 6e Chapter 3 Review Precalculus Complex Numbers 2026-07-29 missed ×2 skipped

Evaluate the expression and write in the form $a + b i$. $(1 - \sqrt{-1})(1 + \sqrt{-1})$

Answer: $2$.

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Stewart Precalc 6e Chapter 3 Review Precalculus Complex Numbers 2026-07-29 missed ×2 skipped

Evaluate the expression and write in the form $a + b i$. $\sqrt{-10} \cdot \sqrt{-40}$

Answer: $-20$.

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Stewart Precalc 6e Chapter 3 Review Precalculus Polynomial Construction 2026-07-29 missed ×2 skipped

Find a polynomial of degree 3 with constant coefficient 12 and zeros $-\dfrac{1}{2}$, $2$, and $3$.

Answer: $P(x) = 4 x^3 - 18 x^2 + 14 x + 12 = 2 (2 x + 1)(x - 2)(x - 3)$.

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Stewart Precalc 6e Chapter 3 Review Precalculus Polynomial Construction 2026-07-29 missed ×2 skipped

Find a polynomial of degree 4 that has integer coefficients and zeros $3 i$ and $4$, with $4$ a double zero.

Answer: $P(x) = x^4 - 8 x^3 + 25 x^2 - 72 x + 144$.

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Stewart Precalc 6e Chapter 3 Review Precalculus Polynomial Construction 2026-07-29 missed ×2 skipped

Does there exist a polynomial of degree 4 with integer coefficients that has zeros $i$, $2 i$, $3 i$, and $4 i$? If so, find it. If not, explain why.

Answer: No such polynomial exists.

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Stewart Precalc 6e Chapter 3 Review Precalculus Polynomial Reasoning 2026-07-29 missed ×2 skipped

Prove that the equation $3 x^4 + 5 x^2 + 2 = 0$ has no real root.

Answer: The equation has no real solution.

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