Algebra 2 Flashcards: Applying The Remainder Theorem

Study Applying The Remainder Theorem in Algebra 2 with focused flashcards that help you recognize the idea, recall the key rule, and apply it in practice-style prompts.

Algebra 2

Applying The Remainder Theorem

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QUESTION

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Find the value of $k$ so that $x-2$ is a factor of $p(x)=x^2+kx-6$ .

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ANSWER

$k=1$ . Set $p(2)=0$ : $4+2k-6=0$ , so $k=1$ .

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What this deck covers

This deck focuses on Applying The Remainder Theorem, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra 2.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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Flashcard 1: Find the value of $k$ so that $x-2$ is a factor of $p(x)=x^2+kx-6$ .

Answer: $k=1$ . Set $p(2)=0$ : $4+2k-6=0$ , so $k=1$ .

Flashcard 2: What is the remainder when dividing $p(x)=x^4-16$ by $x+2$ ?

Answer: $p(-2)=0$ . Calculate $p(-2)=(-2)^4-16=16-16=0$ .

Flashcard 3: What is the remainder when dividing $p(x)=3x^2-12x+9$ by $x-1$ ?

Answer: $p(1)=0$ . Calculate $p(1)=3(1)^2-12(1)+9=3-12+9=0$ .

Flashcard 4: What is the remainder when dividing $p(x)=x^2-1$ by $x- rac{1}{3}$ ?

Answer: $p\left(\frac{1}{3}\right)=-\frac{8}{9}$ . Calculate $p\left(\frac{1}{3}\right)=\left(\frac{1}{3}\right)^2-1=\frac{1}{9}-1=-\frac{8}{9}$ .

Flashcard 5: Identify the remainder when dividing $p(x)=x^2+x+1$ by $x-1$ .

Answer: $p(1)=3$ . Calculate $p(1)=1^2+1+1=1+1+1=3$ .

Flashcard 6: If $p(0)=5$ , what is the remainder when dividing $p(x)$ by $x$ ?

Answer: Remainder $=5$ . Dividing by $x$ means evaluating at $x=0$ .

Flashcard 7: What is the remainder when dividing $p(x)=x^3-2x^2+x-2$ by $x-2$ ?

Answer: $p(2)=0$ . Calculate $p(2)=8-8+2-2=0$ .

Flashcard 8: If the remainder on division by $x-a$ is $r$ , what is $p(a)$ equal to?

Answer: $p(a)=r$ . The remainder equals the polynomial evaluated at $a$ .

Flashcard 9: Identify the remainder when dividing $p(x)=5x^2+2x-3$ by $x-0$ .

Answer: $p(0)=-3$ . Calculate $p(0)=5(0)^2+2(0)-3=-3$ .

Flashcard 10: What is the remainder when dividing $p(x)=x^3-9x$ by $x+3$ ?

Answer: $p(-3)=0$ . Calculate $p(-3)=(-3)^3-9(-3)=-27+27=0$ .

Flashcard 11: What is the remainder when dividing $p(x)=4x^3+1$ by $x- rac{1}{2}$ ?

Answer: $p\left(\frac{1}{2}\right)=\frac{3}{2}$ . Calculate $p\left(\frac{1}{2}\right)=4\left(\frac{1}{2}\right)^3+1=\frac{1}{2}+1=\frac{3}{2}$ .

Flashcard 12: If dividing $p(x)$ by $x-9$ gives remainder $7$ , what is the value of $p(9)$ ?

Answer: $p(9)=7$ . By the Remainder Theorem, $p(a)$ equals the remainder.

Flashcard 13: If $p(6)=-3$ , what is the remainder when dividing $p(x)$ by $x-6$ ?

Answer: Remainder $=-3$ . By the Remainder Theorem, the remainder equals $p(6)$ .

Flashcard 14: Identify the value of $a$ used in the Remainder Theorem when the divisor is $x+7$ .

Answer: $a=-7$ . For divisor $x+7=x-(-7)$ , the value $a=-7$ .

Flashcard 15: Which condition guarantees that $(x-a)$ is a factor of $p(x)$ : $p(a)=0$ or $p(a)\neq 0$ ?

Answer: $p(a)=0$ . When $p(a)=0$ , the remainder is zero, making $(x-a)$ a factor.

Flashcard 16: If dividing $p(x)$ by $x+2$ gives remainder $0$ , what is the value of $p(-2)$ ?

Answer: $p(-2)=0$ . If remainder is 0, then $p(a)=0$ by the theorem.

Flashcard 17: What is the remainder when dividing $p(x)=x^2-2x+2$ by $x+1$ ?

Answer: $p(-1)=5$ . Calculate $p(-1)=(-1)^2-2(-1)+2=1+2+2=5$ .

Flashcard 18: What does the Factor Theorem state using $p(a)$ and the factor $(x-a)$ ?

Answer: $p(a)=0\iff (x-a)$ is a factor of $p(x)$ . The Factor Theorem combines remainder and factorization.

Flashcard 19: Find the remainder when dividing $p(x)=x^3+1$ by $x-1$ .

Answer: $p(1)=2$ . Calculate $p(1)=1^3+1=1+1=2$ .

Flashcard 20: Which divisor corresponds to evaluating $p(-5)$ by the Remainder Theorem: $x-5$ or $x+5$ ?

Answer: $x+5$ . To evaluate $p(-5)$ , divide by $x-(-5)=x+5$ .

Flashcard 21: Which divisor corresponds to evaluating $p(5)$ by the Remainder Theorem: $x-5$ or $x+5$ ?

Answer: $x-5$ . To evaluate $p(5)$ , divide by $x-5$ .

Flashcard 22: If $p(-4)=12$ , what is the remainder when dividing $p(x)$ by $x+4$ ?

Answer: Remainder $=12$ . By the Remainder Theorem, the remainder equals $p(-4)$ .

Flashcard 23: Which statement is true if the remainder on division by $x-4$ is $0$ ?

Answer: $(x-4)$ is a factor of $p(x)$ . A zero remainder means the divisor is a factor.

Flashcard 24: If $(x-a)$ is a factor of $p(x)$ , what must the remainder be when dividing by $(x-a)$ ?

Answer: Remainder $=0$ . If $(x-a)$ is a factor, then $p(a)=0$ .

Flashcard 25: Identify the remainder when dividing $p(x)=x^2+x+1$ by $x+1$ .

Answer: $p(-1)=1$ . Calculate $p(-1)=(-1)^2+(-1)+1=1-1+1=1$ .

Flashcard 26: What is the remainder when dividing $p(x)$ by $x+5$ in terms of $p$ ?

Answer: Remainder $=p(-5)$ . Since $x+5=x-(-5)$ , we have $a=-5$ .

Flashcard 27: Identify the value of $a$ used in the Remainder Theorem when the divisor is $x-7$ .

Answer: $a=7$ . For divisor $x-7$ , the value $a=7$ .

Flashcard 28: Find the value of $k$ so that $x-1$ is a factor of $p(x)=x^3-4x^2+kx+3$ .

Answer: $k=0$ . Set $p(1)=0$ : $1-4+k+3=0$ , so $k=0$ .

Flashcard 29: What is the remainder when dividing $p(x)=x^4-16$ by $x-2$ ?

Answer: $p(2)=0$ . Calculate $p(2)=2^4-16=16-16=0$ .

Flashcard 30: Find the remainder when dividing $p(x)=x^3-1$ by $x+1$ .

Answer: $p(-1)=-2$ . Calculate $p(-1)=(-1)^3-1=-1-1=-2$ .

Flashcard 31: If $p(-2)=0$ , which linear factor must divide $p(x)$ : $x-2$ or $x+2$ ?

Answer: $x+2$ . If $p(-2)=0$ , then $(x+2)$ is a factor.

Flashcard 32: What is the remainder when dividing $p(x)=x^3-4x+1$ by $x-2$ ?

Answer: $p(2)=1$ . Calculate $p(2)=2^3-4(2)+1=8-8+1=1$ .

Flashcard 33: What is the remainder when dividing $p(x)=x^3-2x^2+x-2$ by $x+1$ ?

Answer: $p(-1)=-6$ . Calculate $p(-1)=-1-2-1-2=-6$ .

Flashcard 34: What is the remainder when dividing $p(x)=x^4+x^3-x-1$ by $x+1$ ?

Answer: $p(-1)=0$ . Calculate $p(-1)=(-1)^4+(-1)^3-(-1)-1=1-1+1-1=0$ .

Flashcard 35: What is the remainder when dividing $p(x)=2x^3-x^2+5$ by $x-1$ ?

Answer: $p(1)=6$ . Calculate $p(1)=2(1)^3-(1)^2+5=2-1+5=6$ .

Flashcard 36: What is the remainder when dividing $p(x)=x^3-9x$ by $x-3$ ?

Answer: $p(3)=0$ . Calculate $p(3)=3^3-9(3)=27-27=0$ .

Flashcard 37: If $p(2)=0$ , which linear factor must divide $p(x)$ : $x-2$ or $x+2$ ?

Answer: $x-2$ . If $p(2)=0$ , then $(x-2)$ is a factor.

Flashcard 38: What is the remainder when dividing $p(x)=3x^2-12x+9$ by $x-3$ ?

Answer: $p(3)=0$ . Calculate $p(3)=3(3)^2-12(3)+9=27-36+9=0$ .

Flashcard 39: Find the value of $k$ so that $x+3$ is a factor of $p(x)=2x^2+kx-9$ .

Answer: $k=3$ . Set $p(-3)=0$ : $18-3k-9=0$ , so $k=3$ .

Flashcard 40: Find the value of $k$ so that $x+2$ is a factor of $p(x)=x^3+kx^2-4x+8$ .

Answer: $k=0$ . Set $p(-2)=0$ : $-8+4k+8+8=0$ , so $k=0$ .

Flashcard 41: Find the remainder when dividing $p(x)=x^3+1$ by $x+1$ .

Answer: $p(-1)=0$ . Calculate $p(-1)=(-1)^3+1=-1+1=0$ .

Flashcard 42: What is the remainder when dividing $p(x)=x^4+x^3-x-1$ by $x-1$ ?

Answer: $p(1)=0$ . Calculate $p(1)=1^4+1^3-1-1=1+1-1-1=0$ .

Flashcard 43: What does the Remainder Theorem state for the remainder when dividing $p(x)$ by $x-a$ ?

Answer: Remainder $=p(a)$ . This is the core statement of the Remainder Theorem.

Flashcard 44: What is the remainder when dividing $p(x)=x^2+3x-10$ by $x-2$ ?

Answer: $p(2)=0$ . Calculate $p(2)=2^2+3(2)-10=4+6-10=0$ .

Flashcard 45: What is the remainder when dividing $p(x)=x^3+2x^2-x-2$ by $x+2$ ?

Answer: $p(-2)=0$ . Calculate $p(-2)=(-2)^3+2(-2)^2-(-2)-2=-8+8+2-2=0$ .

Flashcard 46: What is the remainder when dividing $p(x)=2x^3-x^2+5$ by $x+1$ ?

Answer: $p(-1)=2$ . Calculate $p(-1)=2(-1)^3-(-1)^2+5=-2-1+5=2$ .

Flashcard 47: What is the remainder when dividing $p(x)=x^2-2x+2$ by $x-1$ ?

Answer: $p(1)=1$ . Calculate $p(1)=1^2-2(1)+2=1-2+2=1$ .

Flashcard 48: What is the remainder when dividing $p(x)$ by $x-3$ in terms of $p$ ?

Answer: Remainder $=p(3)$ . By the Remainder Theorem, substitute $a=3$ .

Flashcard 49: Find the remainder when dividing $p(x)=x^3-1$ by $x-1$ .

Answer: $p(1)=0$ . Calculate $p(1)=1^3-1=1-1=0$ .

Flashcard 50: What is the remainder when dividing $p(x)=x^2+3x-10$ by $x+5$ ?

Answer: $p(-5)=0$ . Calculate $p(-5)=(-5)^2+3(-5)-10=25-15-10=0$ .

Flashcard 51: What is the remainder when dividing $p(x)=x^3+2x^2-x-2$ by $x-1$ ?

Answer: $p(1)=0$ . Calculate $p(1)=1^3+2(1)^2-1-2=1+2-1-2=0$ .

Algebra 2 Flashcards: Applying The Remainder Theorem

Applying The Remainder Theorem

What this deck covers

How to use these flashcards

All flashcards

Flashcard 1: Find the value of kkk so that x−2x-2x−2 is a factor of p(x)=x2+kx−6p(x)=x^2+kx-6p(x)=x2+kx−6.

Flashcard 2: What is the remainder when dividing p(x)=x4−16p(x)=x^4-16p(x)=x4−16 by x+2x+2x+2?

Flashcard 3: What is the remainder when dividing p(x)=3x2−12x+9p(x)=3x^2-12x+9p(x)=3x2−12x+9 by x−1x-1x−1?

Flashcard 4: What is the remainder when dividing p(x)=x2−1p(x)=x^2-1p(x)=x2−1 by x- rac{1}{3}?

Flashcard 5: Identify the remainder when dividing p(x)=x2+x+1p(x)=x^2+x+1p(x)=x2+x+1 by x−1x-1x−1.

Flashcard 6: If p(0)=5p(0)=5p(0)=5, what is the remainder when dividing p(x)p(x)p(x) by xxx?

Flashcard 7: What is the remainder when dividing p(x)=x3−2x2+x−2p(x)=x^3-2x^2+x-2p(x)=x3−2x2+x−2 by x−2x-2x−2?

Flashcard 8: If the remainder on division by x−ax-ax−a is rrr, what is p(a)p(a)p(a) equal to?

Flashcard 9: Identify the remainder when dividing p(x)=5x2+2x−3p(x)=5x^2+2x-3p(x)=5x2+2x−3 by x−0x-0x−0.

Flashcard 10: What is the remainder when dividing p(x)=x3−9xp(x)=x^3-9xp(x)=x3−9x by x+3x+3x+3?

Flashcard 11: What is the remainder when dividing p(x)=4x3+1p(x)=4x^3+1p(x)=4x3+1 by x- rac{1}{2}?

Flashcard 12: If dividing p(x)p(x)p(x) by x−9x-9x−9 gives remainder 777, what is the value of p(9)p(9)p(9)?

Flashcard 13: If p(6)=−3p(6)=-3p(6)=−3, what is the remainder when dividing p(x)p(x)p(x) by x−6x-6x−6?

Flashcard 14: Identify the value of aaa used in the Remainder Theorem when the divisor is x+7x+7x+7.

Flashcard 15: Which condition guarantees that (x−a)(x-a)(x−a) is a factor of p(x)p(x)p(x): p(a)=0p(a)=0p(a)=0 or p(a)≠0p(a)\neq 0p(a)=0?

Flashcard 16: If dividing p(x)p(x)p(x) by x+2x+2x+2 gives remainder 000, what is the value of p(−2)p(-2)p(−2)?

Flashcard 17: What is the remainder when dividing p(x)=x2−2x+2p(x)=x^2-2x+2p(x)=x2−2x+2 by x+1x+1x+1?

Flashcard 18: What does the Factor Theorem state using p(a)p(a)p(a) and the factor (x−a)(x-a)(x−a)?

Flashcard 19: Find the remainder when dividing p(x)=x3+1p(x)=x^3+1p(x)=x3+1 by x−1x-1x−1.

Flashcard 20: Which divisor corresponds to evaluating p(−5)p(-5)p(−5) by the Remainder Theorem: x−5x-5x−5 or x+5x+5x+5?

Flashcard 21: Which divisor corresponds to evaluating p(5)p(5)p(5) by the Remainder Theorem: x−5x-5x−5 or x+5x+5x+5?

Flashcard 22: If p(−4)=12p(-4)=12p(−4)=12, what is the remainder when dividing p(x)p(x)p(x) by x+4x+4x+4?

Flashcard 23: Which statement is true if the remainder on division by x−4x-4x−4 is 000?

Flashcard 24: If (x−a)(x-a)(x−a) is a factor of p(x)p(x)p(x), what must the remainder be when dividing by (x−a)(x-a)(x−a)?

Flashcard 25: Identify the remainder when dividing p(x)=x2+x+1p(x)=x^2+x+1p(x)=x2+x+1 by x+1x+1x+1.

Flashcard 26: What is the remainder when dividing p(x)p(x)p(x) by x+5x+5x+5 in terms of ppp?

Flashcard 27: Identify the value of aaa used in the Remainder Theorem when the divisor is x−7x-7x−7.

Flashcard 28: Find the value of kkk so that x−1x-1x−1 is a factor of p(x)=x3−4x2+kx+3p(x)=x^3-4x^2+kx+3p(x)=x3−4x2+kx+3.

Flashcard 29: What is the remainder when dividing p(x)=x4−16p(x)=x^4-16p(x)=x4−16 by x−2x-2x−2?

Flashcard 30: Find the remainder when dividing p(x)=x3−1p(x)=x^3-1p(x)=x3−1 by x+1x+1x+1.

Flashcard 31: If p(−2)=0p(-2)=0p(−2)=0, which linear factor must divide p(x)p(x)p(x): x−2x-2x−2 or x+2x+2x+2?

Flashcard 32: What is the remainder when dividing p(x)=x3−4x+1p(x)=x^3-4x+1p(x)=x3−4x+1 by x−2x-2x−2?

Flashcard 33: What is the remainder when dividing p(x)=x3−2x2+x−2p(x)=x^3-2x^2+x-2p(x)=x3−2x2+x−2 by x+1x+1x+1?

Flashcard 34: What is the remainder when dividing p(x)=x4+x3−x−1p(x)=x^4+x^3-x-1p(x)=x4+x3−x−1 by x+1x+1x+1?

Flashcard 35: What is the remainder when dividing p(x)=2x3−x2+5p(x)=2x^3-x^2+5p(x)=2x3−x2+5 by x−1x-1x−1?

Flashcard 36: What is the remainder when dividing p(x)=x3−9xp(x)=x^3-9xp(x)=x3−9x by x−3x-3x−3?

Flashcard 37: If p(2)=0p(2)=0p(2)=0, which linear factor must divide p(x)p(x)p(x): x−2x-2x−2 or x+2x+2x+2?

Flashcard 38: What is the remainder when dividing p(x)=3x2−12x+9p(x)=3x^2-12x+9p(x)=3x2−12x+9 by x−3x-3x−3?

Flashcard 39: Find the value of kkk so that x+3x+3x+3 is a factor of p(x)=2x2+kx−9p(x)=2x^2+kx-9p(x)=2x2+kx−9.

Flashcard 40: Find the value of kkk so that x+2x+2x+2 is a factor of p(x)=x3+kx2−4x+8p(x)=x^3+kx^2-4x+8p(x)=x3+kx2−4x+8.

Flashcard 41: Find the remainder when dividing p(x)=x3+1p(x)=x^3+1p(x)=x3+1 by x+1x+1x+1.

Flashcard 42: What is the remainder when dividing p(x)=x4+x3−x−1p(x)=x^4+x^3-x-1p(x)=x4+x3−x−1 by x−1x-1x−1?

Flashcard 43: What does the Remainder Theorem state for the remainder when dividing p(x)p(x)p(x) by x−ax-ax−a?

Flashcard 44: What is the remainder when dividing p(x)=x2+3x−10p(x)=x^2+3x-10p(x)=x2+3x−10 by x−2x-2x−2?

Flashcard 45: What is the remainder when dividing p(x)=x3+2x2−x−2p(x)=x^3+2x^2-x-2p(x)=x3+2x2−x−2 by x+2x+2x+2?

Flashcard 46: What is the remainder when dividing p(x)=2x3−x2+5p(x)=2x^3-x^2+5p(x)=2x3−x2+5 by x+1x+1x+1?

Flashcard 47: What is the remainder when dividing p(x)=x2−2x+2p(x)=x^2-2x+2p(x)=x2−2x+2 by x−1x-1x−1?

Flashcard 48: What is the remainder when dividing p(x)p(x)p(x) by x−3x-3x−3 in terms of ppp?

Flashcard 49: Find the remainder when dividing p(x)=x3−1p(x)=x^3-1p(x)=x3−1 by x−1x-1x−1.

Flashcard 50: What is the remainder when dividing p(x)=x2+3x−10p(x)=x^2+3x-10p(x)=x2+3x−10 by x+5x+5x+5?

Flashcard 51: What is the remainder when dividing p(x)=x3+2x2−x−2p(x)=x^3+2x^2-x-2p(x)=x3+2x2−x−2 by x−1x-1x−1?