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Stewart Precalc 6e Chapter 3 Review
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Topic Breakdown
Finding All Zeros
Weak
0/10 · 0%
Complex Numbers
Weak
0/9 · 0%
Real Zeros and Graph
Weak
0/8 · 0%
Polynomial Graph Transformation
Weak
0/6 · 0%
Rational Function Graphing
Weak
0/6 · 0%
Quadratic Function - Standard Form
Weak
0/4 · 0%
Graphing to Solve Equations
Weak
0/4 · 0%
Rational Function Asymptotes
Weak
0/4 · 0%
Remainder Theorem
Weak
0/3 · 0%
Polynomial Construction
Weak
0/3 · 0%
Quadratic Function - Maximum/Minimum
Weak
0/2 · 0%
Factor Theorem
Weak
0/2 · 0%
Rational Zeros and Descartes' Rule
Weak
0/2 · 0%
Factoring with Complex
Weak
0/2 · 0%
Quadratic Application - Projectile Motion
Weak
0/1 · 0%
Quadratic Application - Profit Maximization
Weak
0/1 · 0%
Polynomial Reasoning
Weak
0/1 · 0%
Polynomial Intersections
Weak
0/1 · 0%
Target your weak topics with focused practice.
Practice weak topics →Results by Question
1
Quadratic Function - Standard Form
Wrong
A quadratic function is given. (a) Express the function in standard form. (b) Graph the function. \(f(x) = x^2 + 4 x + 1\).
(No answer submitted)
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No explanation
2
Quadratic Function - Standard Form
Wrong
A quadratic function is given. (a) Express the function in standard form. (b) Graph the function. \(f(x) = -2 x^2 + 12 x + 12\).
(No answer submitted)
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No explanation
3
Quadratic Function - Standard Form
Wrong
A quadratic function is given. (a) Express the function in standard form. (b) Graph the function. \(g(x) = 1 + 8 x - x^2\).
(No answer submitted)
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No explanation
4
Quadratic Function - Standard Form
Wrong
A quadratic function is given. (a) Express the function in standard form. (b) Graph the function. \(g(x) = 6 x - 3 x^2\).
(No answer submitted)
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No explanation
5
Quadratic Function - Maximum/Minimum
Wrong
Find the maximum or minimum value of the quadratic function \(f(x) = 2 x^2 + 4 x - 5\).
(No answer submitted)
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No explanation
6
Quadratic Function - Maximum/Minimum
Wrong
Find the maximum or minimum value of the quadratic function \(g(x) = 1 - x - x^2\).
(No answer submitted)
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No explanation
7
Quadratic Application - Projectile Motion
Wrong
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?
(No answer submitted)
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No explanation
8
Quadratic Application - Profit Maximization
Wrong
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?
(No answer submitted)
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No explanation
9
Polynomial Graph Transformation
Wrong
Graph the polynomial by transforming an appropriate graph of the form \(y = x^n\). Show clearly all \(x\)- and \(y\)-intercepts. \(P(x) = -x^3 + 64\).
(No answer submitted)
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No explanation
10
Polynomial Graph Transformation
Wrong
Graph the polynomial by transforming an appropriate graph of the form \(y = x^n\). Show clearly all \(x\)- and \(y\)-intercepts. \(P(x) = 2 x^3 - 16\).
(No answer submitted)
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No explanation
11
Polynomial Graph Transformation
Wrong
Graph the polynomial by transforming an appropriate graph of the form \(y = x^n\). Show clearly all \(x\)- and \(y\)-intercepts. \(P(x) = 2 (x + 1)^4 - 32\).
(No answer submitted)
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No explanation
12
Polynomial Graph Transformation
Wrong
Graph the polynomial by transforming an appropriate graph of the form \(y = x^n\). Show clearly all \(x\)- and \(y\)-intercepts. \(P(x) = 81 - (x - 3)^4\).
(No answer submitted)
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No explanation
13
Polynomial Graph Transformation
Wrong
Graph the polynomial by transforming an appropriate graph of the form \(y = x^n\). Show clearly all \(x\)- and \(y\)-intercepts. \(P(x) = 32 + (x - 1)^5\).
(No answer submitted)
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No explanation
14
Polynomial Graph Transformation
Wrong
Graph the polynomial by transforming an appropriate graph of the form \(y = x^n\). Show clearly all \(x\)- and \(y\)-intercepts. \(P(x) = -3 (x + 2)^5 + 96\).
(No answer submitted)
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No explanation
15
Remainder Theorem
Wrong
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 \(Q(-3)\)
(No answer submitted)
Answer
\(Q(-3) = 21\).
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Explanation
Synthetic division by \(x + 3\) on \(1, 4, 7, 10, 15\): \(1\); \(4 - 3 = 1\); \(7 - 3 = 4\); \(10 - 12 = -2\); \(15 + 6 = 21\).
16
Factor Theorem
Wrong
Show that \(\dfrac{1}{2}\) is a zero of the polynomial
\(P(x) = 2 x^4 + x^3 - 5 x^2 + 10 x - 4\)
(No answer submitted)
Answer
\(P\left(\dfrac{1}{2}\right) = 0\).
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Explanation
\(P\left(\dfrac{1}{2}\right) = 2 \cdot \dfrac{1}{16} + \dfrac{1}{8} - 5 \cdot \dfrac{1}{4} + 10 \cdot \dfrac{1}{2} - 4 = \dfrac{1}{8} + \dfrac{1}{8} - \dfrac{5}{4} + 5 - 4 = \dfrac{1}{4} - \dfrac{5}{4} + 1 = -1 + 1 = 0\).
17
Factor Theorem
Wrong
Use the Factor Theorem to show that \(x + 4\) is a factor of the polynomial
\(P(x) = x^5 + 4 x^4 - 7 x^3 - 23 x^2 + 23 x + 12\)
(No answer submitted)
Answer
\(P(-4) = 0\), so \(x + 4\) is a factor.
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Explanation
\(P(-4) = -1024 + 4 \cdot 256 - 7 \cdot (-64) - 23 \cdot 16 - 92 + 12 = -1024 + 1024 + 448 - 368 - 92 + 12 = 0\). By the Factor Theorem, \(x + 4\) divides \(P(x)\).
18
Remainder Theorem
Wrong
What is the remainder when the polynomial
\(P(x) = x^{500} + 6 x^{201} - x^2 - 2 x + 4\)
is divided by \(x - 1\)?
(No answer submitted)
Answer
Remainder \(= 8\).
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Explanation
By the Remainder Theorem, the remainder is \(P(1) = 1 + 6 - 1 - 2 + 4 = 8\).
19
Remainder Theorem
Wrong
What is the remainder when \(x^{101} - x^4 + 2\) is divided by \(x + 1\)?
(No answer submitted)
Answer
Remainder \(= 0\).
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Explanation
By the Remainder Theorem, evaluate at \(x = -1\): \((-1)^{101} - (-1)^4 + 2 = -1 - 1 + 2 = 0\).
20
Rational Zeros and Descartes' Rule
Wrong
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.
\(P(x) = x^5 - 6 x^3 - x^2 + 2 x + 18\)
(No answer submitted)
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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Explanation
Possible rational zeros are \(\pm \dfrac{p}{q}\) where \(p\) divides the constant \(18\) and \(q\) divides the leading coefficient \(1\). Signs of \(P(x)\): \(+, -, -, +, +\) give 2 sign changes. Signs of \(P(-x) = -x^5 + 6 x^3 - x^2 - 2 x + 18\): \(-, +, -, -, +\) give 3 sign changes.
21
Rational Zeros and Descartes' Rule
Wrong
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.
\(P(x) = 6 x^4 + 3 x^3 + x^2 + 3 x + 4\)
(No answer submitted)
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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Explanation
All coefficients of \(P(x)\) are positive (no sign changes) so no positive real zeros. \(P(-x) = 6 x^4 - 3 x^3 + x^2 - 3 x + 4\) has signs \(+, -, +, -, +\) with 4 sign changes.
22
Real Zeros and Graph
Wrong
A polynomial \(P\) is given. (a) Find all real zeros of \(P\), and state their multiplicities. (b) Sketch the graph of \(P\).
\(P(x) = x^3 - 16 x\)
(No answer submitted)
Answer
Zeros: \(0\), \(4\), \(-4\) (each multiplicity 1). \(P(x) = x (x - 4)(x + 4)\).
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Explanation
\(P(x) = x (x^2 - 16) = x (x - 4)(x + 4)\). Three simple real zeros.
23
Real Zeros and Graph
Wrong
A polynomial \(P\) is given. (a) Find all real zeros of \(P\), and state their multiplicities. (b) Sketch the graph of \(P\).
\(P(x) = x^3 - 3 x^2 - 4 x\)
(No answer submitted)
Answer
Zeros: \(0\), \(4\), \(-1\) (each multiplicity 1). \(P(x) = x (x - 4)(x + 1)\).
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Explanation
\(P(x) = x (x^2 - 3 x - 4) = x (x - 4)(x + 1)\).
24
Real Zeros and Graph
Wrong
A polynomial \(P\) is given. (a) Find all real zeros of \(P\), and state their multiplicities. (b) Sketch the graph of \(P\).
\(P(x) = x^4 + x^3 - 2 x^2\)
(No answer submitted)
Answer
Zeros: \(0\) (multiplicity 2), \(-2\) (multiplicity 1), \(1\) (multiplicity 1).
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Explanation
\(P(x) = x^2 (x^2 + x - 2) = x^2 (x + 2)(x - 1)\). The zero at \(0\) has multiplicity 2 (graph touches); the others are simple.
25
Real Zeros and Graph
Wrong
A polynomial \(P\) is given. (a) Find all real zeros of \(P\), and state their multiplicities. (b) Sketch the graph of \(P\).
\(P(x) = x^4 - 5 x^2 + 4\)
(No answer submitted)
Answer
Zeros: \(\pm 1\), \(\pm 2\) (each multiplicity 1).
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Explanation
Let \(y = x^2\): \(y^2 - 5 y + 4 = (y - 1)(y - 4)\). So \(x^2 = 1\) or \(x^2 = 4\), giving \(x = \pm 1, \pm 2\). Factored: \((x - 1)(x + 1)(x - 2)(x + 2)\).
26
Real Zeros and Graph
Wrong
A polynomial \(P\) is given. (a) Find all real zeros of \(P\), and state their multiplicities. (b) Sketch the graph of \(P\).
\(P(x) = x^4 - 2 x^3 - 7 x^2 + 8 x + 12\)
(No answer submitted)
Answer
Zeros: \(-1\), \(-2\), \(2\), \(3\) (each multiplicity 1). \(P(x) = (x + 1)(x + 2)(x - 2)(x - 3)\).
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Explanation
Testing rational candidates, \(P(-1) = 0\). Dividing by \(x + 1\) gives \(x^3 - 3 x^2 - 4 x + 12\). \(P(3) = 0\), then dividing yields \(x^2 - 4 = (x - 2)(x + 2)\).
27
Real Zeros and Graph
Wrong
A polynomial \(P\) is given. (a) Find all real zeros of \(P\), and state their multiplicities. (b) Sketch the graph of \(P\).
\(P(x) = x^4 - 2 x^3 - 2 x^2 + 8 x - 8\)
(No answer submitted)
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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Explanation
\(P(2) = 0\) and dividing gives \(x^3 - 2 x + 4\); \(P(-2)\) on the cubic gives \(0\), dividing yields \(x^2 - 2 x + 2\), which has complex roots.
28
Real Zeros and Graph
Wrong
A polynomial \(P\) is given. (a) Find all real zeros of \(P\), and state their multiplicities. (b) Sketch the graph of \(P\).
\(P(x) = 2 x^4 + x^3 + 2 x^2 - 3 x - 2\)
(No answer submitted)
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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Explanation
\(P(1) = 0\), dividing leaves \(2 x^3 + 3 x^2 + 5 x + 2\). Testing \(x = -\dfrac{1}{2}\) gives \(0\); dividing yields \(2 x^2 + 2 x + 4\), whose discriminant is negative.
29
Real Zeros and Graph
Wrong
A polynomial \(P\) is given. (a) Find all real zeros of \(P\), and state their multiplicities. (b) Sketch the graph of \(P\).
\(P(x) = 9 x^5 - 21 x^4 + 10 x^3 + 6 x^2 - 3 x - 1\)
(No answer submitted)
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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Explanation
Three successive synthetic divisions by \(x - 1\) leave \(9 x^2 - 6 x - 3 = 3 (3 x + 1)(x - 1)\), which contributes one more \((x - 1)\) factor and one \((3 x + 1)\) factor. Combined exponents give \((x - 1)^3\) and \(\left(x + \dfrac{1}{3}\right)^2\).
30
Complex Numbers
Wrong
Evaluate the expression and write in the form \(a + b i\).
\((2 - 3 i) + (1 + 4 i)\)
(No answer submitted)
Answer
\(3 + i\).
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Explanation
Sum real parts: \(2 + 1 = 3\). Sum imaginary parts: \(-3 + 4 = 1\).
31
Complex Numbers
Wrong
Evaluate the expression and write in the form \(a + b i\).
\((3 - 6 i) - (6 - 4 i)\)
(No answer submitted)
Answer
\(-3 - 2 i\).
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Explanation
\((3 - 6) + (-6 - (-4)) i = -3 - 2 i\).
32
Complex Numbers
Wrong
Evaluate the expression and write in the form \(a + b i\).
\((2 + i)(3 - 2 i)\)
(No answer submitted)
Answer
\(8 - i\).
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Explanation
FOIL: \(6 - 4 i + 3 i - 2 i^2 = 6 - i + 2 = 8 - i\).
33
Complex Numbers
Wrong
Evaluate the expression and write in the form \(a + b i\).
\(4 i \left(2 - \dfrac{1}{2} i\right)\)
(No answer submitted)
Answer
\(2 + 8 i\).
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Explanation
\(4 i \cdot 2 - 4 i \cdot \dfrac{1}{2} i = 8 i - 2 i^2 = 8 i + 2 = 2 + 8 i\).
34
Complex Numbers
Wrong
Evaluate the expression and write in the form \(a + b i\).
\(\dfrac{4 + 2 i}{2 - i}\)
(No answer submitted)
Answer
\(\dfrac{6}{5} + \dfrac{8}{5} i\).
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Explanation
Multiply numerator and denominator by the conjugate \(2 + i\): numerator \(= (4 + 2 i)(2 + i) = 8 + 4 i + 4 i + 2 i^2 = 6 + 8 i\); denominator \(= 4 + 1 = 5\).
35
Complex Numbers
Wrong
Evaluate the expression and write in the form \(a + b i\).
\(\dfrac{8 + 3 i}{4 + 3 i}\)
(No answer submitted)
Answer
\(\dfrac{41}{25} - \dfrac{12}{25} i\).
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Explanation
Multiply by conjugate \(4 - 3 i\): numerator \(= 32 - 24 i + 12 i - 9 i^2 = 41 - 12 i\); denominator \(= 16 + 9 = 25\).
36
Complex Numbers
Wrong
Evaluate the expression and write in the form \(a + b i\).
\((1 + i)^3\)
(No answer submitted)
Answer
\(-2 + 2 i\).
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Explanation
\((1 + i)^2 = 1 + 2 i + i^2 = 2 i\), so \((1 + i)^3 = (1 + i)(2 i) = 2 i + 2 i^2 = -2 + 2 i\).
37
Complex Numbers
Wrong
Evaluate the expression and write in the form \(a + b i\).
\((1 - \sqrt{-1})(1 + \sqrt{-1})\)
(No answer submitted)
Answer
\(2\).
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Explanation
\(\sqrt{-1} = i\), so \((1 - i)(1 + i) = 1 - i^2 = 1 - (-1) = 2\).
38
Complex Numbers
Wrong
Evaluate the expression and write in the form \(a + b i\).
\(\sqrt{-10} \cdot \sqrt{-40}\)
(No answer submitted)
Answer
\(-20\).
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Explanation
Convert first: \(\sqrt{-10} = i \sqrt{10}\) and \(\sqrt{-40} = 2 i \sqrt{10}\). Product \(= 2 i^2 \cdot 10 = -20\). Caution: \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a b}\) fails when both are negative.
39
Polynomial Construction
Wrong
Find a polynomial of degree 3 with constant coefficient 12 and zeros \(-\dfrac{1}{2}\), \(2\), and \(3\).
(No answer submitted)
Answer
\(P(x) = 4 x^3 - 18 x^2 + 14 x + 12 = 2 (2 x + 1)(x - 2)(x - 3)\).
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Explanation
Start with \(a \left(x + \dfrac{1}{2}\right)(x - 2)(x - 3)\). Setting \(P(0) = a \cdot \dfrac{1}{2} \cdot (-2)(-3) = 3 a = 12\) gives \(a = 4\). Then \(4 \left(x + \dfrac{1}{2}\right)(x - 2)(x - 3) = 2 (2 x + 1)(x - 2)(x - 3)\).
40
Polynomial Construction
Wrong
Find a polynomial of degree 4 that has integer coefficients and zeros \(3 i\) and \(4\), with \(4\) a double zero.
(No answer submitted)
Answer
\(P(x) = x^4 - 8 x^3 + 25 x^2 - 72 x + 144\).
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Explanation
Since coefficients are real (integer), \(3 i\) forces conjugate \(-3 i\). So \(P(x) = (x - 3 i)(x + 3 i)(x - 4)^2 = (x^2 + 9)(x^2 - 8 x + 16) = x^4 - 8 x^3 + 25 x^2 - 72 x + 144\).
41
Polynomial Construction
Wrong
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.
(No answer submitted)
Answer
No such polynomial exists.
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Explanation
A polynomial with integer (hence real) coefficients must have complex zeros occurring in conjugate pairs. The given zeros \(i, 2 i, 3 i, 4 i\) are not paired with their conjugates \(-i, -2 i, -3 i, -4 i\), so all eight conjugates would be required, making the minimum degree 8.
42
Polynomial Reasoning
Wrong
Prove that the equation \(3 x^4 + 5 x^2 + 2 = 0\) has no real root.
(No answer submitted)
Answer
The equation has no real solution.
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Explanation
Let \(y = x^2 \geq 0\) for real \(x\). The equation becomes \(3 y^2 + 5 y + 2 = (3 y + 2)(y + 1) = 0\), giving \(y = -\dfrac{2}{3}\) or \(y = -1\). Both are negative, so \(x^2\) cannot equal them; there are no real solutions.
43
Finding All Zeros
Wrong
\( P(x) = x^3 - 3 x^2 - 13 x + 15 \)
(No answer submitted)
Answer
Zeros: \(1\), \(5\), \(-3\) (each multiplicity 1). \(P(x) = (x - 1)(x - 5)(x + 3)\).
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Explanation
\(P(1) = 0\). Dividing by \(x - 1\) gives \(x^2 - 2 x - 15 = (x - 5)(x + 3)\).
44
Finding All Zeros
Wrong
\( P(x) = 2 x^3 + 5 x^2 - 6 x - 9 \)
(No answer submitted)
Answer
Zeros: \(-1\), \(\dfrac{3}{2}\), \(-3\) (each multiplicity 1). \(P(x) = (x + 1)(2 x - 3)(x + 3)\).
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Explanation
\(P(-1) = 0\). Dividing yields \(2 x^2 + 3 x - 9 = (2 x - 3)(x + 3)\).
45
Finding All Zeros
Wrong
\( P(x) = x^4 + 6 x^3 + 17 x^2 + 28 x + 20 \)
(No answer submitted)
Answer
Zeros: \(-2\) (multiplicity 2), \(-1 \pm 2 i\). \(P(x) = (x + 2)^2 (x^2 + 2 x + 5)\).
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Explanation
\(P(-2) = 0\) twice (two successive divisions by \(x + 2\) leave \(x^2 + 2 x + 5\), which has complex roots \(-1 \pm 2 i\)).
46
Finding All Zeros
Wrong
\( P(x) = x^4 + 7 x^3 + 9 x^2 - 17 x - 20 \)
(No answer submitted)
Answer
Zeros: \(-1\), \(-4\), \(-1 \pm \sqrt{6}\) (each multiplicity 1). \(P(x) = (x + 1)(x + 4)(x^2 + 2 x - 5)\).
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Explanation
\(P(-1) = 0\), then \(P(-4) = 0\) on the cubic. Remaining factor \(x^2 + 2 x - 5\) has roots \(-1 \pm \sqrt{6}\).
47
Finding All Zeros
Wrong
\( P(x) = x^5 - 3 x^4 - x^3 + 11 x^2 - 12 x + 4 \)
(No answer submitted)
Answer
Zeros: \(1\) (multiplicity 3), \(2\), \(-2\). \(P(x) = (x - 1)^3 (x - 2)(x + 2)\).
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Explanation
Three successive divisions by \(x - 1\) leave \(x^2 - 4 = (x - 2)(x + 2)\).
48
Finding All Zeros
Wrong
\( P(x) = x^4 - 81 \)
(No answer submitted)
Answer
Zeros: \(3\), \(-3\), \(3 i\), \(-3 i\). \(P(x) = (x - 3)(x + 3)(x^2 + 9)\).
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Explanation
\(x^4 - 81 = (x^2 - 9)(x^2 + 9) = (x - 3)(x + 3)(x^2 + 9)\). The factor \(x^2 + 9\) gives complex zeros \(\pm 3 i\).
49
Finding All Zeros
Wrong
\( P(x) = x^6 - 64 \)
(No answer submitted)
Answer
Zeros: \(2\), \(-2\), \(1 \pm i \sqrt{3}\), \(-1 \pm i \sqrt{3}\). \(P(x) = (x - 2)(x + 2)(x^2 - 2 x + 4)(x^2 + 2 x + 4)\).
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Explanation
Factor as a difference of cubes (in \(x^2\)) twice: \(x^6 - 64 = (x^3 - 8)(x^3 + 8) = (x - 2)(x^2 + 2 x + 4)(x + 2)(x^2 - 2 x + 4)\). Each irreducible quadratic gives a complex conjugate pair.
50
Finding All Zeros
Wrong
\( P(x) = 18 x^3 + 3 x^2 - 4 x - 1 \)
(No answer submitted)
Answer
Zeros: \(\dfrac{1}{2}\) (multiplicity 1), \(-\dfrac{1}{3}\) (multiplicity 2). \(P(x) = 2 (2 x - 1)(3 x + 1)^2\).
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Explanation
Testing \(x = \dfrac{1}{2}\) gives \(0\). Dividing leaves \(18 x^2 + 12 x + 2 = 2 (9 x^2 + 6 x + 1) = 2 (3 x + 1)^2\).
51
Finding All Zeros
Wrong
\( P(x) = 6 x^4 - 18 x^3 + 6 x^2 - 30 x + 36 \)
(No answer submitted)
Answer
Zeros: \(1\), \(3\), \(\dfrac{-1 \pm i \sqrt{7}}{2}\). \(P(x) = 6 (x - 1)(x - 3)(x^2 + x + 2)\).
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Explanation
\(P(1) = 0\), dividing leaves \(6 x^3 - 12 x^2 - 6 x - 36 = 6 (x^3 - 2 x^2 - x - 6)\); testing \(x = 3\) gives \(0\); remaining factor \(x^2 + x + 2\) has discriminant \(-7\).
52
Finding All Zeros
Wrong
\( P(x) = x^4 + 15 x^2 + 54 \)
(No answer submitted)
Answer
Zeros: \(\pm i \sqrt{6}\), \(\pm 3 i\). \(P(x) = (x^2 + 6)(x^2 + 9)\).
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Explanation
Substituting \(y = x^2\): \(y^2 + 15 y + 54 = (y + 6)(y + 9)\). So \(x^2 = -6\) or \(x^2 = -9\), giving the four imaginary zeros.
53
Graphing to Solve Equations
Wrong
Use a graphing device to find all real solutions of the equation.
\(2 x^2 = 5 x + 3\)
(No answer submitted)
Answer
\(x = 3\), \(x = -\dfrac{1}{2}\).
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Explanation
Rewriting as \(2 x^2 - 5 x - 3 = 0\) and factoring: \((2 x + 1)(x - 3) = 0\).
54
Graphing to Solve Equations
Wrong
Use a graphing device to find all real solutions of the equation.
\(x^3 + x^2 - 14 x - 24 = 0\)
(No answer submitted)
Answer
\(x = -2\), \(x = 4\), \(x = -3\).
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Explanation
Testing rational candidates: \(P(-2) = 0\). Dividing by \(x + 2\) leaves \(x^2 - x - 12 = (x - 4)(x + 3)\).
55
Graphing to Solve Equations
Wrong
Use a graphing device to find all real solutions of the equation.
\(x^4 - 3 x^3 - 3 x^2 - 9 x - 2 = 0\)
(No answer submitted)
Answer
Two real solutions: \(x \approx -0.24\) and \(x \approx 4.24\).
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Explanation
Descartes' Rule of Signs gives one positive and one or three negative zeros. Graphing reveals exactly two real zeros (the other two are complex). Approximate using a calculator.
56
Graphing to Solve Equations
Wrong
Use a graphing device to find all real solutions of the equation.
\(x^5 = x + 3\)
(No answer submitted)
Answer
One real solution: \(x \approx 1.34\).
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Explanation
Rewrite as \(x^5 - x - 3 = 0\). The derivative \(5 x^4 - 1\) has only two real roots, and analysis shows exactly one real root for the quintic, located between 1 and 2.
57
Factoring with Complex
Wrong
A polynomial function \(P\) is given. Find all the real zeros of \(P\), and factor \(P\) completely into linear and irreducible quadratic factors with real coefficients.
\(P(x) = x^3 - 2 x - 4\)
(No answer submitted)
Answer
Real zero: \(x = 2\). \(P(x) = (x - 2)(x^2 + 2 x + 2)\).
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Explanation
\(P(2) = 0\), dividing leaves \(x^2 + 2 x + 2\), discriminant \(4 - 8 = -4 < 0\), so this quadratic is irreducible over the reals.
58
Factoring with Complex
Wrong
A polynomial function \(P\) is given. Find all the real zeros of \(P\), and factor \(P\) completely into linear and irreducible quadratic factors with real coefficients.
\(P(x) = x^4 + 3 x^2 - 4\)
(No answer submitted)
Answer
Real zeros: \(x = 1\) and \(x = -1\). \(P(x) = (x - 1)(x + 1)(x^2 + 4)\).
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Explanation
Let \(y = x^2\): \(y^2 + 3 y - 4 = (y - 1)(y + 4)\). So \(x^2 = 1\) (real) or \(x^2 = -4\) (complex). The factor \(x^2 + 4\) is irreducible over \(RR\).
59
Rational Function Graphing
Wrong
Graph the rational function. Show clearly all \(x\)- and \(y\)-intercepts and asymptotes.
\(r(x) = \dfrac{3 x - 12}{x + 1}\)
(No answer submitted)
Answer
\(x\)-intercept: \(4\); \(y\)-intercept: \(-12\); vertical asymptote: \(x = -1\); horizontal asymptote: \(y = 3\).
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Explanation
Setting numerator \(= 0\): \(3 x - 12 = 0\), so \(x = 4\). \(r(0) = -12\). Denominator zero at \(x = -1\). Degrees equal: HA at ratio of leading coefficients \(= 3\).
60
Rational Function Graphing
Wrong
Graph the rational function. Show clearly all \(x\)- and \(y\)-intercepts and asymptotes.
\(r(x) = \dfrac{1}{(x + 2)^2}\)
(No answer submitted)
Answer
No \(x\)-intercept; \(y\)-intercept: \(\dfrac{1}{4}\); vertical asymptote: \(x = -2\); horizontal asymptote: \(y = 0\).
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Explanation
Numerator never zero, so no \(x\)-intercept. \(r(0) = \dfrac{1}{4}\). Vertical asymptote at \(x = -2\) (double, function approaches \(+\infty\) on both sides). HA at \(y = 0\) since denominator dominates.
61
Rational Function Graphing
Wrong
Graph the rational function. Show clearly all \(x\)- and \(y\)-intercepts and asymptotes.
\(r(x) = \dfrac{x - 2}{x^2 - 2 x - 8}\)
(No answer submitted)
Answer
\(x\)-intercept: \(2\); \(y\)-intercept: \(\dfrac{1}{4}\); vertical asymptotes: \(x = 4\) and \(x = -2\); horizontal asymptote: \(y = 0\).
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Explanation
Denominator factors: \(x^2 - 2 x - 8 = (x - 4)(x + 2)\). Numerator zero at \(x = 2\). \(r(0) = \dfrac{-2}{-8} = \dfrac{1}{4}\). Degree of denominator exceeds numerator, so HA is \(y = 0\).
62
Rational Function Graphing
Wrong
Graph the rational function. Show clearly all \(x\)- and \(y\)-intercepts and asymptotes.
\(r(x) = \dfrac{2 x^2 - 6 x - 7}{x - 4}\)
(No answer submitted)
Answer
\(x\)-intercepts: \(x = \dfrac{3 \pm \sqrt{23}}{2}\); \(y\)-intercept: \(\dfrac{7}{4}\); vertical asymptote: \(x = 4\); slant asymptote: \(y = 2 x + 2\).
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Explanation
Long division: \(r(x) = 2 x + 2 + \dfrac{1}{x - 4}\), giving slant asymptote \(y = 2 x + 2\). Solving \(2 x^2 - 6 x - 7 = 0\) with the quadratic formula gives the \(x\)-intercepts.
63
Rational Function Graphing
Wrong
Graph the rational function. Show clearly all \(x\)- and \(y\)-intercepts and asymptotes.
\(r(x) = \dfrac{x^2 - 9}{2 x^2 + 1}\)
(No answer submitted)
Answer
\(x\)-intercepts: \(\pm 3\); \(y\)-intercept: \(-9\); no vertical asymptote; horizontal asymptote: \(y = \dfrac{1}{2}\).
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Explanation
Numerator zero at \(x = \pm 3\). Denominator \(2 x^2 + 1 > 0\) always, so no VA. Degrees equal; HA at ratio of leading coefficients \(= \dfrac{1}{2}\).
64
Rational Function Graphing
Wrong
Graph the rational function. Show clearly all \(x\)- and \(y\)-intercepts and asymptotes.
\(r(x) = \dfrac{x^3 + 27}{x + 4}\)
(No answer submitted)
Answer
\(x\)-intercept: \(-3\); \(y\)-intercept: \(\dfrac{27}{4}\); vertical asymptote: \(x = -4\); end behavior like \(y = x^2 - 4 x + 16\).
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Explanation
\(x^3 + 27 = (x + 3)(x^2 - 3 x + 9)\) gives one real root \(x = -3\). Long division: \(r(x) = x^2 - 4 x + 16 - \dfrac{37}{x + 4}\), so end behavior matches the quadratic \(y = x^2 - 4 x + 16\).
65
Rational Function Asymptotes
Wrong
Use a graphing device to analyze the graph of the rational function. Find all \(x\)- and \(y\)-intercepts and all vertical, horizontal, and slant asymptotes. If the function has no horizontal or slant asymptote, find a polynomial that has the same end behavior as the rational function.
\(r(x) = \dfrac{x - 3}{2 x + 6}\)
(No answer submitted)
Answer
\(x\)-intercept: \(3\); \(y\)-intercept: \(-\dfrac{1}{2}\); vertical asymptote: \(x = -3\); horizontal asymptote: \(y = \dfrac{1}{2}\).
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Explanation
Numerator zero at \(x = 3\); denominator zero at \(x = -3\). \(r(0) = \dfrac{-3}{6} = -\dfrac{1}{2}\). Equal degrees give HA equal to ratio of leading coefficients \(\dfrac{1}{2}\).
66
Rational Function Asymptotes
Wrong
Use a graphing device to analyze the graph of the rational function. Find all \(x\)- and \(y\)-intercepts and all vertical, horizontal, and slant asymptotes. If the function has no horizontal or slant asymptote, find a polynomial that has the same end behavior as the rational function.
\(r(x) = \dfrac{2 x - 7}{x^2 + 9}\)
(No answer submitted)
Answer
\(x\)-intercept: \(\dfrac{7}{2}\); \(y\)-intercept: \(-\dfrac{7}{9}\); no vertical asymptote; horizontal asymptote: \(y = 0\).
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Explanation
Denominator \(x^2 + 9 > 0\) always. Numerator dominates by lower degree, so HA at \(y = 0\).
67
Rational Function Asymptotes
Wrong
Use a graphing device to analyze the graph of the rational function. Find all \(x\)- and \(y\)-intercepts and all vertical, horizontal, and slant asymptotes.
\(r(x) = \dfrac{x^3 + 8}{x^2 - x - 2}\)
(No answer submitted)
Answer
\(x\)-intercept: \(-2\); \(y\)-intercept: \(-4\); vertical asymptotes: \(x = 2\) and \(x = -1\); slant asymptote: \(y = x + 1\).
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Explanation
Denominator factors \((x - 2)(x + 1)\). Long division gives \(r(x) = x + 1 + \dfrac{3 x + 10}{x^2 - x - 2}\), hence slant asymptote \(y = x + 1\).
68
Rational Function Asymptotes
Wrong
Use a graphing device to analyze the graph of the rational function. Find all \(x\)- and \(y\)-intercepts and all vertical, horizontal, and slant asymptotes. If the function has no horizontal or slant asymptote, find a polynomial that has the same end behavior as the rational function.
\(r(x) = \dfrac{2 x^3 - x^2}{x + 1}\)
(No answer submitted)
Answer
\(x\)-intercepts: \(0\) (double) and \(\dfrac{1}{2}\); \(y\)-intercept: \(0\); vertical asymptote: \(x = -1\); end behavior like \(y = 2 x^2 - 3 x + 3\).
Compare with the answer above and grade yourself:
Explanation
Numerator \(= x^2 (2 x - 1)\). Long division: \(r(x) = 2 x^2 - 3 x + 3 - \dfrac{3}{x + 1}\), so end behavior matches the quadratic \(y = 2 x^2 - 3 x + 3\).
69
Polynomial Intersections
Wrong
Find the coordinates of all points of intersection of the graphs of
\(y = x^4 + x^2 + 24 x\) and \(y = 6 x^3 + 20\)
(No answer submitted)
Answer
\((1, 26)\), \((2, 68)\), \((-2, -28)\), \((5, 770)\).
Compare with the answer above and grade yourself:
Explanation
Setting equal: \(x^4 - 6 x^3 + x^2 + 24 x - 20 = 0\). Testing \(x = 1\) gives \(0\); dividing leaves \(x^3 - 5 x^2 - 4 x + 20\). \(x = 5\) gives \(0\); dividing leaves \(x^2 - 4 = (x - 2)(x + 2)\). Substitute each \(x\) back into either curve to get \(y\).
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| # | Date | Score | Accuracy | |
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| Current | 2026-07-29 03:31 | 0 / 69 | 0% | |
| 2 | 2026-07-23 07:02 | 0 / 69 | 0% | View |