This algebra 2 polynomial test answer key gives the full solution and the scoring guide for each of the 9 test questions on polynomial division and the remainder theorem.
This answer key shows a full solution and a scoring guide for each of the 9 test questions. Award each point as described, and give partial credit when the method is correct but the arithmetic has a slip.
Algebra 2 Polynomial Test Answer Key: questions 1–5
Question 1: Long division (2 points)
\(x^{3}: x=x^{2}\); subtract \(x^{3}-x^{2}\) to get \(3x^{2}-6x\). \(3x^{2}: x=3x\); subtract \(3x^{2}-3x\) to get \(-3x+5\). \(-3x: x=-3\); subtract \(-3x+3\) to get \(2\).
Quotient \(x^{2}+3x-3\), remainder \(2\).
Scoring: 1 point for the correct quotient, 1 point for the correct remainder.
Question 2: Division with a missing term (2 points)
Write the dividend as \(x^{4}+0x^{3}-5x^{2}+3x-2\) and use \(c=-2\): \(1\); \(0-2=-2\); \(-5+4=-1\); \(3+2=5\); \(-2-10=-12\).
Quotient \(x^{3}-2x^{2}-x+5\), remainder \(-12\).
Scoring: 1 point for including the zero coefficient and using \(c=-2\); 1 point for the correct quotient and remainder.
Question 3: Remainder Theorem (2 points)
a) \(p(3)=2(27)-9-21+6=54-9-21+6=30\).
b) \(p(-2)=-16-4+14+6=0\), so by the Factor Theorem \(x+2\) is a factor of \(p(x)\) and \(-2\) is a zero.
a) \(30\) b) \(x+2\) is a factor
Scoring: 1 point for each part.
Question 4: Factor Theorem and zeros (3 points)
a) \(f(2)=8-4-28+24=0\), so \(x-2\) is a factor.
b) Synthetic division with \(c=2\): \(1\); \(-1+2=1\); \(-14+2=-12\); \(24-24=0\). The quotient is \(x^{2}+x-12=(x+4)(x-3)\).
c) The zeros are \(-4\), \(2\), and \(3\).
a) \(f(2)=0\) b) \(f(x)=(x-2)(x+4)(x-3)\) c) \(-4,\ 2,\ 3\)
Scoring: 1 point per part. In part b, 1 point requires all three factors correct.
Question 5: Find k and factor (2 points)
a) \(p(-3)=-27+9k+15+3=9k-9=0\), so \(k=1\).
b) With \(k=1\), \(p(x)=x^{3}+x^{2}-5x+3\). Divide by \(x+3\) with \(c=-3\): \(1\); \(1-3=-2\); \(-5+6=1\); \(3-3=0\). The quotient is \(x^{2}-2x+1=(x-1)^{2}\).
a) \(k=1\) b) \((x+3)(x-1)^{2}\)
Scoring: 1 point for each part.
Algebra 2 Polynomial Test Answer Key: questions 6–9
Question 6: Function from a graph (2 points)
a) The zeros are \(-1\), \(2\), and \(4\), so \(f(x)=(x+1)(x-2)(x-4)\). Check the y-intercept: \(f(0)=(1)(-2)(-4)=8\).
b) \(f(5)=(6)(3)(1)=18\).
a) \(f(x)=(x+1)(x-2)(x-4)\) b) \(18\)
Scoring: 1 point for each part.
Question 7: Polynomial identity (2 points)
a) \((x^{2}+y^{2})^{2}=x^{4}+2x^{2}y^{2}+y^{4}\). Also \((x^{2}-y^{2})^{2}+(2xy)^{2}=x^{4}-2x^{2}y^{2}+y^{4}+4x^{2}y^{2}=x^{4}+2x^{2}y^{2}+y^{4}\). The sides are equal.
b) \(x^{2}-y^{2}=15\), \(2xy=8\), \(x^{2}+y^{2}=17\). Check: \(225+64=289=17^{2}\).
b) legs \(15\) and \(8\), hypotenuse \(17\)
Scoring: 1 point for a correct expansion of both sides; 1 point for the three side lengths.
Question 8: Dimensions of a box (3 points)
a) The base area is \((x+5)(x+2)=x^{2}+7x+10\). Divide: \(x^{3}: x^{2}=x\); subtract \(x^{3}+7x^{2}+10x\) to get \(3x^{2}+21x+30\); \(3x^{2}: x^{2}=3\); subtract \(3x^{2}+21x+30\) to get \(0\). So \(h=x+3\).
b) For \(x=4\): \(9\cdot6\cdot7=378\) cubic inches. Check: \(V(4)=64+160+124+30=378\).
a) \(h=x+3\) inches b) \(378\) cubic inches
Scoring: part a, 1 point for the base area and 1 point for the division and answer; part b, 1 point.
Question 9: Divide by 2x plus 1 (2 points)
\(6x^{3}: 2x=3x^{2}\); subtract \(6x^{3}+3x^{2}\) to get \(-4x^{2}+6x\). \(-4x^{2}: 2x=-2x\); subtract \(-4x^{2}-2x\) to get \(8x+7\). \(8x: 2x=4\); subtract \(8x+4\) to get \(3\).
Quotient \(3x^{2}-2x+4\), remainder \(3\).
Scoring: 1 point for the correct quotient, 1 point for the correct remainder.
Scoring summary
20 points: add the points of every question of this algebra 2 polynomial test answer key.
How to use this algebra 2 polynomial test answer key
Use this algebra 2 polynomial test answer key to score the test fairly and to see which skills need more practice.
See every chapter on the Algebra 2 math page.