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Stewart Precalc 6e Chapter 3 Review
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단원별 정답률
Finding All Zeros
약점
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Complex Numbers
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Real Zeros and Graph
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Polynomial Graph Transformation
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Rational Function Graphing
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Quadratic Function - Standard Form
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Graphing to Solve Equations
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Rational Function Asymptotes
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Remainder Theorem
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Polynomial Construction
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Quadratic Function - Maximum/Minimum
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Factor Theorem
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Rational Zeros and Descartes' Rule
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Factoring with Complex
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Quadratic Application - Projectile Motion
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Quadratic Application - Profit Maximization
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Polynomial Reasoning
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Polynomial Intersections
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1
Quadratic Function - Standard Form
오답
A quadratic function is given. (a) Express the function in standard form. (b) Graph the function. \(f(x) = x^2 + 4 x + 1\).
(미작성)
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2
Quadratic Function - Standard Form
오답
A quadratic function is given. (a) Express the function in standard form. (b) Graph the function. \(f(x) = -2 x^2 + 12 x + 12\).
(미작성)
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3
Quadratic Function - Standard Form
오답
A quadratic function is given. (a) Express the function in standard form. (b) Graph the function. \(g(x) = 1 + 8 x - x^2\).
(미작성)
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4
Quadratic Function - Standard Form
오답
A quadratic function is given. (a) Express the function in standard form. (b) Graph the function. \(g(x) = 6 x - 3 x^2\).
(미작성)
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5
Quadratic Function - Maximum/Minimum
오답
Find the maximum or minimum value of the quadratic function \(f(x) = 2 x^2 + 4 x - 5\).
(미작성)
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6
Quadratic Function - Maximum/Minimum
오답
Find the maximum or minimum value of the quadratic function \(g(x) = 1 - x - x^2\).
(미작성)
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7
Quadratic Application - Projectile Motion
오답
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?
(미작성)
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8
Quadratic Application - Profit Maximization
오답
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?
(미작성)
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9
Polynomial Graph Transformation
오답
Graph the polynomial by transforming an appropriate graph of the form \(y = x^n\). Show clearly all \(x\)- and \(y\)-intercepts. \(P(x) = -x^3 + 64\).
(미작성)
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10
Polynomial Graph Transformation
오답
Graph the polynomial by transforming an appropriate graph of the form \(y = x^n\). Show clearly all \(x\)- and \(y\)-intercepts. \(P(x) = 2 x^3 - 16\).
(미작성)
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11
Polynomial Graph Transformation
오답
Graph the polynomial by transforming an appropriate graph of the form \(y = x^n\). Show clearly all \(x\)- and \(y\)-intercepts. \(P(x) = 2 (x + 1)^4 - 32\).
(미작성)
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12
Polynomial Graph Transformation
오답
Graph the polynomial by transforming an appropriate graph of the form \(y = x^n\). Show clearly all \(x\)- and \(y\)-intercepts. \(P(x) = 81 - (x - 3)^4\).
(미작성)
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13
Polynomial Graph Transformation
오답
Graph the polynomial by transforming an appropriate graph of the form \(y = x^n\). Show clearly all \(x\)- and \(y\)-intercepts. \(P(x) = 32 + (x - 1)^5\).
(미작성)
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14
Polynomial Graph Transformation
오답
Graph the polynomial by transforming an appropriate graph of the form \(y = x^n\). Show clearly all \(x\)- and \(y\)-intercepts. \(P(x) = -3 (x + 2)^5 + 96\).
(미작성)
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15
Remainder Theorem
오답
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 \(Q(-3)\)
(미작성)
정답
\(Q(-3) = 21\).
위 정답과 비교하여 채점하세요:
해설
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
오답
Show that \(\dfrac{1}{2}\) is a zero of the polynomial
\(P(x) = 2 x^4 + x^3 - 5 x^2 + 10 x - 4\)
(미작성)
정답
\(P\left(\dfrac{1}{2}\right) = 0\).
위 정답과 비교하여 채점하세요:
해설
\(P\left(\dfrac{1}{2}\right) = 2 \cdot \dfrac{1}{16} + \dfrac{1}{8} - 5 \cdot \dfrac{1}{4} + 10 \cdot \dfrac{1}{2} - 4 = \dfrac{1}{8} + \dfrac{1}{8} - \dfrac{5}{4} + 5 - 4 = \dfrac{1}{4} - \dfrac{5}{4} + 1 = -1 + 1 = 0\).
17
Factor Theorem
오답
Use the Factor Theorem to show that \(x + 4\) is a factor of the polynomial
\(P(x) = x^5 + 4 x^4 - 7 x^3 - 23 x^2 + 23 x + 12\)
(미작성)
정답
\(P(-4) = 0\), so \(x + 4\) is a factor.
위 정답과 비교하여 채점하세요:
해설
\(P(-4) = -1024 + 4 \cdot 256 - 7 \cdot (-64) - 23 \cdot 16 - 92 + 12 = -1024 + 1024 + 448 - 368 - 92 + 12 = 0\). By the Factor Theorem, \(x + 4\) divides \(P(x)\).
18
Remainder Theorem
오답
What is the remainder when the polynomial
\(P(x) = x^{500} + 6 x^{201} - x^2 - 2 x + 4\)
is divided by \(x - 1\)?
(미작성)
정답
Remainder \(= 8\).
위 정답과 비교하여 채점하세요:
해설
By the Remainder Theorem, the remainder is \(P(1) = 1 + 6 - 1 - 2 + 4 = 8\).
19
Remainder Theorem
오답
What is the remainder when \(x^{101} - x^4 + 2\) is divided by \(x + 1\)?
(미작성)
정답
Remainder \(= 0\).
위 정답과 비교하여 채점하세요:
해설
By the Remainder Theorem, evaluate at \(x = -1\): \((-1)^{101} - (-1)^4 + 2 = -1 - 1 + 2 = 0\).
20
Rational Zeros and Descartes' Rule
오답
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.
\(P(x) = x^5 - 6 x^3 - x^2 + 2 x + 18\)
(미작성)
정답
(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.
위 정답과 비교하여 채점하세요:
해설
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
오답
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.
\(P(x) = 6 x^4 + 3 x^3 + x^2 + 3 x + 4\)
(미작성)
정답
(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.
위 정답과 비교하여 채점하세요:
해설
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
오답
A polynomial \(P\) is given. (a) Find all real zeros of \(P\), and state their multiplicities. (b) Sketch the graph of \(P\).
\(P(x) = x^3 - 16 x\)
(미작성)
정답
Zeros: \(0\), \(4\), \(-4\) (each multiplicity 1). \(P(x) = x (x - 4)(x + 4)\).
위 정답과 비교하여 채점하세요:
해설
\(P(x) = x (x^2 - 16) = x (x - 4)(x + 4)\). Three simple real zeros.
23
Real Zeros and Graph
오답
A polynomial \(P\) is given. (a) Find all real zeros of \(P\), and state their multiplicities. (b) Sketch the graph of \(P\).
\(P(x) = x^3 - 3 x^2 - 4 x\)
(미작성)
정답
Zeros: \(0\), \(4\), \(-1\) (each multiplicity 1). \(P(x) = x (x - 4)(x + 1)\).
위 정답과 비교하여 채점하세요:
해설
\(P(x) = x (x^2 - 3 x - 4) = x (x - 4)(x + 1)\).
24
Real Zeros and Graph
오답
A polynomial \(P\) is given. (a) Find all real zeros of \(P\), and state their multiplicities. (b) Sketch the graph of \(P\).
\(P(x) = x^4 + x^3 - 2 x^2\)
(미작성)
정답
Zeros: \(0\) (multiplicity 2), \(-2\) (multiplicity 1), \(1\) (multiplicity 1).
위 정답과 비교하여 채점하세요:
해설
\(P(x) = x^2 (x^2 + x - 2) = x^2 (x + 2)(x - 1)\). The zero at \(0\) has multiplicity 2 (graph touches); the others are simple.
25
Real Zeros and Graph
오답
A polynomial \(P\) is given. (a) Find all real zeros of \(P\), and state their multiplicities. (b) Sketch the graph of \(P\).
\(P(x) = x^4 - 5 x^2 + 4\)
(미작성)
정답
Zeros: \(\pm 1\), \(\pm 2\) (each multiplicity 1).
위 정답과 비교하여 채점하세요:
해설
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
오답
A polynomial \(P\) is given. (a) Find all real zeros of \(P\), and state their multiplicities. (b) Sketch the graph of \(P\).
\(P(x) = x^4 - 2 x^3 - 7 x^2 + 8 x + 12\)
(미작성)
정답
Zeros: \(-1\), \(-2\), \(2\), \(3\) (each multiplicity 1). \(P(x) = (x + 1)(x + 2)(x - 2)(x - 3)\).
위 정답과 비교하여 채점하세요:
해설
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
오답
A polynomial \(P\) is given. (a) Find all real zeros of \(P\), and state their multiplicities. (b) Sketch the graph of \(P\).
\(P(x) = x^4 - 2 x^3 - 2 x^2 + 8 x - 8\)
(미작성)
정답
Real zeros: \(2\), \(-2\) (each multiplicity 1). Complex zeros: \(1 \pm i\). \(P(x) = (x - 2)(x + 2)(x^2 - 2 x + 2)\).
위 정답과 비교하여 채점하세요:
해설
\(P(2) = 0\) 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
오답
A polynomial \(P\) is given. (a) Find all real zeros of \(P\), and state their multiplicities. (b) Sketch the graph of \(P\).
\(P(x) = 2 x^4 + x^3 + 2 x^2 - 3 x - 2\)
(미작성)
정답
Real zeros: \(1\) and \(-\dfrac{1}{2}\) (each multiplicity 1). Remaining factor: \(2 x^2 + 2 x + 4\) (complex roots).
위 정답과 비교하여 채점하세요:
해설
\(P(1) = 0\), 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
오답
A polynomial \(P\) is given. (a) Find all real zeros of \(P\), and state their multiplicities. (b) Sketch the graph of \(P\).
\(P(x) = 9 x^5 - 21 x^4 + 10 x^3 + 6 x^2 - 3 x - 1\)
(미작성)
정답
Real zeros: \(1\) (multiplicity 3), \(-\dfrac{1}{3}\) (multiplicity 2). \(P(x) = 9 (x - 1)^3 \left(x + \dfrac{1}{3}\right)^2\).
위 정답과 비교하여 채점하세요:
해설
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
오답
Evaluate the expression and write in the form \(a + b i\).
\((2 - 3 i) + (1 + 4 i)\)
(미작성)
정답
\(3 + i\).
위 정답과 비교하여 채점하세요:
해설
Sum real parts: \(2 + 1 = 3\). Sum imaginary parts: \(-3 + 4 = 1\).
31
Complex Numbers
오답
Evaluate the expression and write in the form \(a + b i\).
\((3 - 6 i) - (6 - 4 i)\)
(미작성)
정답
\(-3 - 2 i\).
위 정답과 비교하여 채점하세요:
해설
\((3 - 6) + (-6 - (-4)) i = -3 - 2 i\).
32
Complex Numbers
오답
Evaluate the expression and write in the form \(a + b i\).
\((2 + i)(3 - 2 i)\)
(미작성)
정답
\(8 - i\).
위 정답과 비교하여 채점하세요:
해설
FOIL: \(6 - 4 i + 3 i - 2 i^2 = 6 - i + 2 = 8 - i\).
33
Complex Numbers
오답
Evaluate the expression and write in the form \(a + b i\).
\(4 i \left(2 - \dfrac{1}{2} i\right)\)
(미작성)
정답
\(2 + 8 i\).
위 정답과 비교하여 채점하세요:
해설
\(4 i \cdot 2 - 4 i \cdot \dfrac{1}{2} i = 8 i - 2 i^2 = 8 i + 2 = 2 + 8 i\).
34
Complex Numbers
오답
Evaluate the expression and write in the form \(a + b i\).
\(\dfrac{4 + 2 i}{2 - i}\)
(미작성)
정답
\(\dfrac{6}{5} + \dfrac{8}{5} i\).
위 정답과 비교하여 채점하세요:
해설
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
오답
Evaluate the expression and write in the form \(a + b i\).
\(\dfrac{8 + 3 i}{4 + 3 i}\)
(미작성)
정답
\(\dfrac{41}{25} - \dfrac{12}{25} i\).
위 정답과 비교하여 채점하세요:
해설
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
오답
Evaluate the expression and write in the form \(a + b i\).
\((1 + i)^3\)
(미작성)
정답
\(-2 + 2 i\).
위 정답과 비교하여 채점하세요:
해설
\((1 + i)^2 = 1 + 2 i + i^2 = 2 i\), so \((1 + i)^3 = (1 + i)(2 i) = 2 i + 2 i^2 = -2 + 2 i\).
37
Complex Numbers
오답
Evaluate the expression and write in the form \(a + b i\).
\((1 - \sqrt{-1})(1 + \sqrt{-1})\)
(미작성)
정답
\(2\).
위 정답과 비교하여 채점하세요:
해설
\(\sqrt{-1} = i\), so \((1 - i)(1 + i) = 1 - i^2 = 1 - (-1) = 2\).
38
Complex Numbers
오답
Evaluate the expression and write in the form \(a + b i\).
\(\sqrt{-10} \cdot \sqrt{-40}\)
(미작성)
정답
\(-20\).
위 정답과 비교하여 채점하세요:
해설
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
오답
Find a polynomial of degree 3 with constant coefficient 12 and zeros \(-\dfrac{1}{2}\), \(2\), and \(3\).
(미작성)
정답
\(P(x) = 4 x^3 - 18 x^2 + 14 x + 12 = 2 (2 x + 1)(x - 2)(x - 3)\).
위 정답과 비교하여 채점하세요:
해설
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
오답
Find a polynomial of degree 4 that has integer coefficients and zeros \(3 i\) and \(4\), with \(4\) a double zero.
(미작성)
정답
\(P(x) = x^4 - 8 x^3 + 25 x^2 - 72 x + 144\).
위 정답과 비교하여 채점하세요:
해설
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
오답
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.
(미작성)
정답
No such polynomial exists.
위 정답과 비교하여 채점하세요:
해설
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
오답
Prove that the equation \(3 x^4 + 5 x^2 + 2 = 0\) has no real root.
(미작성)
정답
The equation has no real solution.
위 정답과 비교하여 채점하세요:
해설
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
오답
\( P(x) = x^3 - 3 x^2 - 13 x + 15 \)
(미작성)
정답
Zeros: \(1\), \(5\), \(-3\) (each multiplicity 1). \(P(x) = (x - 1)(x - 5)(x + 3)\).
위 정답과 비교하여 채점하세요:
해설
\(P(1) = 0\). Dividing by \(x - 1\) gives \(x^2 - 2 x - 15 = (x - 5)(x + 3)\).
44
Finding All Zeros
오답
\( P(x) = 2 x^3 + 5 x^2 - 6 x - 9 \)
(미작성)
정답
Zeros: \(-1\), \(\dfrac{3}{2}\), \(-3\) (each multiplicity 1). \(P(x) = (x + 1)(2 x - 3)(x + 3)\).
위 정답과 비교하여 채점하세요:
해설
\(P(-1) = 0\). Dividing yields \(2 x^2 + 3 x - 9 = (2 x - 3)(x + 3)\).
45
Finding All Zeros
오답
\( P(x) = x^4 + 6 x^3 + 17 x^2 + 28 x + 20 \)
(미작성)
정답
Zeros: \(-2\) (multiplicity 2), \(-1 \pm 2 i\). \(P(x) = (x + 2)^2 (x^2 + 2 x + 5)\).
위 정답과 비교하여 채점하세요:
해설
\(P(-2) = 0\) 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
오답
\( P(x) = x^4 + 7 x^3 + 9 x^2 - 17 x - 20 \)
(미작성)
정답
Zeros: \(-1\), \(-4\), \(-1 \pm \sqrt{6}\) (each multiplicity 1). \(P(x) = (x + 1)(x + 4)(x^2 + 2 x - 5)\).
위 정답과 비교하여 채점하세요:
해설
\(P(-1) = 0\), then \(P(-4) = 0\) on the cubic. Remaining factor \(x^2 + 2 x - 5\) has roots \(-1 \pm \sqrt{6}\).
47
Finding All Zeros
오답
\( P(x) = x^5 - 3 x^4 - x^3 + 11 x^2 - 12 x + 4 \)
(미작성)
정답
Zeros: \(1\) (multiplicity 3), \(2\), \(-2\). \(P(x) = (x - 1)^3 (x - 2)(x + 2)\).
위 정답과 비교하여 채점하세요:
해설
Three successive divisions by \(x - 1\) leave \(x^2 - 4 = (x - 2)(x + 2)\).
48
Finding All Zeros
오답
\( P(x) = x^4 - 81 \)
(미작성)
정답
Zeros: \(3\), \(-3\), \(3 i\), \(-3 i\). \(P(x) = (x - 3)(x + 3)(x^2 + 9)\).
위 정답과 비교하여 채점하세요:
해설
\(x^4 - 81 = (x^2 - 9)(x^2 + 9) = (x - 3)(x + 3)(x^2 + 9)\). The factor \(x^2 + 9\) gives complex zeros \(\pm 3 i\).
49
Finding All Zeros
오답
\( P(x) = x^6 - 64 \)
(미작성)
정답
Zeros: \(2\), \(-2\), \(1 \pm i \sqrt{3}\), \(-1 \pm i \sqrt{3}\). \(P(x) = (x - 2)(x + 2)(x^2 - 2 x + 4)(x^2 + 2 x + 4)\).
위 정답과 비교하여 채점하세요:
해설
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
오답
\( P(x) = 18 x^3 + 3 x^2 - 4 x - 1 \)
(미작성)
정답
Zeros: \(\dfrac{1}{2}\) (multiplicity 1), \(-\dfrac{1}{3}\) (multiplicity 2). \(P(x) = 2 (2 x - 1)(3 x + 1)^2\).
위 정답과 비교하여 채점하세요:
해설
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
오답
\( P(x) = 6 x^4 - 18 x^3 + 6 x^2 - 30 x + 36 \)
(미작성)
정답
Zeros: \(1\), \(3\), \(\dfrac{-1 \pm i \sqrt{7}}{2}\). \(P(x) = 6 (x - 1)(x - 3)(x^2 + x + 2)\).
위 정답과 비교하여 채점하세요:
해설
\(P(1) = 0\), 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
오답
\( P(x) = x^4 + 15 x^2 + 54 \)
(미작성)
정답
Zeros: \(\pm i \sqrt{6}\), \(\pm 3 i\). \(P(x) = (x^2 + 6)(x^2 + 9)\).
위 정답과 비교하여 채점하세요:
해설
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
오답
Use a graphing device to find all real solutions of the equation.
\(2 x^2 = 5 x + 3\)
(미작성)
정답
\(x = 3\), \(x = -\dfrac{1}{2}\).
위 정답과 비교하여 채점하세요:
해설
Rewriting as \(2 x^2 - 5 x - 3 = 0\) and factoring: \((2 x + 1)(x - 3) = 0\).
54
Graphing to Solve Equations
오답
Use a graphing device to find all real solutions of the equation.
\(x^3 + x^2 - 14 x - 24 = 0\)
(미작성)
정답
\(x = -2\), \(x = 4\), \(x = -3\).
위 정답과 비교하여 채점하세요:
해설
Testing rational candidates: \(P(-2) = 0\). Dividing by \(x + 2\) leaves \(x^2 - x - 12 = (x - 4)(x + 3)\).
55
Graphing to Solve Equations
오답
Use a graphing device to find all real solutions of the equation.
\(x^4 - 3 x^3 - 3 x^2 - 9 x - 2 = 0\)
(미작성)
정답
Two real solutions: \(x \approx -0.24\) and \(x \approx 4.24\).
위 정답과 비교하여 채점하세요:
해설
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
오답
Use a graphing device to find all real solutions of the equation.
\(x^5 = x + 3\)
(미작성)
정답
One real solution: \(x \approx 1.34\).
위 정답과 비교하여 채점하세요:
해설
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
오답
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.
\(P(x) = x^3 - 2 x - 4\)
(미작성)
정답
Real zero: \(x = 2\). \(P(x) = (x - 2)(x^2 + 2 x + 2)\).
위 정답과 비교하여 채점하세요:
해설
\(P(2) = 0\), dividing leaves \(x^2 + 2 x + 2\), discriminant \(4 - 8 = -4 < 0\), so this quadratic is irreducible over the reals.
58
Factoring with Complex
오답
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.
\(P(x) = x^4 + 3 x^2 - 4\)
(미작성)
정답
Real zeros: \(x = 1\) and \(x = -1\). \(P(x) = (x - 1)(x + 1)(x^2 + 4)\).
위 정답과 비교하여 채점하세요:
해설
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
오답
Graph the rational function. Show clearly all \(x\)- and \(y\)-intercepts and asymptotes.
\(r(x) = \dfrac{3 x - 12}{x + 1}\)
(미작성)
정답
\(x\)-intercept: \(4\); \(y\)-intercept: \(-12\); vertical asymptote: \(x = -1\); horizontal asymptote: \(y = 3\).
위 정답과 비교하여 채점하세요:
해설
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
오답
Graph the rational function. Show clearly all \(x\)- and \(y\)-intercepts and asymptotes.
\(r(x) = \dfrac{1}{(x + 2)^2}\)
(미작성)
정답
No \(x\)-intercept; \(y\)-intercept: \(\dfrac{1}{4}\); vertical asymptote: \(x = -2\); horizontal asymptote: \(y = 0\).
위 정답과 비교하여 채점하세요:
해설
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
오답
Graph the rational function. Show clearly all \(x\)- and \(y\)-intercepts and asymptotes.
\(r(x) = \dfrac{x - 2}{x^2 - 2 x - 8}\)
(미작성)
정답
\(x\)-intercept: \(2\); \(y\)-intercept: \(\dfrac{1}{4}\); vertical asymptotes: \(x = 4\) and \(x = -2\); horizontal asymptote: \(y = 0\).
위 정답과 비교하여 채점하세요:
해설
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
오답
Graph the rational function. Show clearly all \(x\)- and \(y\)-intercepts and asymptotes.
\(r(x) = \dfrac{2 x^2 - 6 x - 7}{x - 4}\)
(미작성)
정답
\(x\)-intercepts: \(x = \dfrac{3 \pm \sqrt{23}}{2}\); \(y\)-intercept: \(\dfrac{7}{4}\); vertical asymptote: \(x = 4\); slant asymptote: \(y = 2 x + 2\).
위 정답과 비교하여 채점하세요:
해설
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
오답
Graph the rational function. Show clearly all \(x\)- and \(y\)-intercepts and asymptotes.
\(r(x) = \dfrac{x^2 - 9}{2 x^2 + 1}\)
(미작성)
정답
\(x\)-intercepts: \(\pm 3\); \(y\)-intercept: \(-9\); no vertical asymptote; horizontal asymptote: \(y = \dfrac{1}{2}\).
위 정답과 비교하여 채점하세요:
해설
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
오답
Graph the rational function. Show clearly all \(x\)- and \(y\)-intercepts and asymptotes.
\(r(x) = \dfrac{x^3 + 27}{x + 4}\)
(미작성)
정답
\(x\)-intercept: \(-3\); \(y\)-intercept: \(\dfrac{27}{4}\); vertical asymptote: \(x = -4\); end behavior like \(y = x^2 - 4 x + 16\).
위 정답과 비교하여 채점하세요:
해설
\(x^3 + 27 = (x + 3)(x^2 - 3 x + 9)\) 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
오답
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.
\(r(x) = \dfrac{x - 3}{2 x + 6}\)
(미작성)
정답
\(x\)-intercept: \(3\); \(y\)-intercept: \(-\dfrac{1}{2}\); vertical asymptote: \(x = -3\); horizontal asymptote: \(y = \dfrac{1}{2}\).
위 정답과 비교하여 채점하세요:
해설
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
오답
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.
\(r(x) = \dfrac{2 x - 7}{x^2 + 9}\)
(미작성)
정답
\(x\)-intercept: \(\dfrac{7}{2}\); \(y\)-intercept: \(-\dfrac{7}{9}\); no vertical asymptote; horizontal asymptote: \(y = 0\).
위 정답과 비교하여 채점하세요:
해설
Denominator \(x^2 + 9 > 0\) always. Numerator dominates by lower degree, so HA at \(y = 0\).
67
Rational Function Asymptotes
오답
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.
\(r(x) = \dfrac{x^3 + 8}{x^2 - x - 2}\)
(미작성)
정답
\(x\)-intercept: \(-2\); \(y\)-intercept: \(-4\); vertical asymptotes: \(x = 2\) and \(x = -1\); slant asymptote: \(y = x + 1\).
위 정답과 비교하여 채점하세요:
해설
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
오답
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.
\(r(x) = \dfrac{2 x^3 - x^2}{x + 1}\)
(미작성)
정답
\(x\)-intercepts: \(0\) (double) and \(\dfrac{1}{2}\); \(y\)-intercept: \(0\); vertical asymptote: \(x = -1\); end behavior like \(y = 2 x^2 - 3 x + 3\).
위 정답과 비교하여 채점하세요:
해설
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
오답
Find the coordinates of all points of intersection of the graphs of
\(y = x^4 + x^2 + 24 x\) and \(y = 6 x^3 + 20\)
(미작성)
정답
\((1, 26)\), \((2, 68)\), \((-2, -28)\), \((5, 770)\).
위 정답과 비교하여 채점하세요:
해설
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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| # | 날짜 | 점수 | 정답률 | |
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