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Algebra • Polynomials & Higher Degree Roots Practice Drills (59 Qs)• Official Answer Keys

CAT Polynomials Practice Questions (55+ Questions)

Remainder Theorem, Factor Theorem, Vieta Relations for Cubics, and Polynomial Graphs

59 Total Questions
MCQ: 41 (+3 / -1)
TITA: 18 (0 Negative Penalty)
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Core Formulas & Shortcut Matrix: Polynomials & Higher Degree Roots Practice Drills

Remainder TheoremFormula #1
P(x) ÷ (x - a) \implies \text{Remainder} = P(a)

Fundamental theorem for evaluating polynomial remainders.

Vieta for Cubic ax^3 + bx^2 + cx + d = 0Formula #2
\sum \alpha = - ba\frac{b}{a} , \quad \sum \alpha\beta = ca\frac{c}{a} , \quad \alpha\beta\gamma = - da\frac{d}{a}$

Symmetric root relations.

Exam Hall Traps & Speedbreakers to Avoid
  • •Forgetting that if degree of divisor is k, maximum degree of remainder is k - 1.

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Official Exam Questions & Explanations (25 of 59)

Sorted in official convenor sequence
Question 1 of 59
ThinkCAT Practice SetQAAlgebra • PolynomialsEasy
If $f(x) = x^2 - 5x + 6 , what is the sum of the roots of $f(x) = 0$?
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Official Correct Answer: A. By Vieta's formulas, the sum of the roots of a quadratic equation $ax^2 + bx + c = 0$ is - b/a$. For $f(x) = x^2 - 5x + 6 , a = 1$ and $b = -5$. Therefore, the sum of the roots is $-(-5)/1 = 5$. Hence, A is correct.
Question 2 of 59
ThinkCAT Practice SetQAAlgebra • PolynomialsModerate
If $p(x) = x^3 - 3x^2 + 2x , what is the value of $p(1)$?
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Official Correct Answer: A. Substitute $x = 1$ into $p(x)$. Thus, $p(1) = 1^3 - 3(1^2) + 2(1) = 1 - 3 + 2 = 0$. Hence, A is correct.
Question 3 of 59
ThinkCAT Practice SetQAAlgebra • PolynomialsHard
If $q(x) = x^4 - 4x^3 + 6x^2 - 4x + 1 , what is the value of $q(1 + i)$?
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Official Correct Answer: A. Notice that $q(x)$ is a binomial expansion of $(x-1)^4$. Thus, $q(1 + i) = (1 + i - 1)^4 = i^4 = 1$. Since $i^4 = 1 , q(1 + i) = 0$. Hence, A is correct.
Question 4 of 59
ThinkCAT Practice SetQAAlgebra • PolynomialsEasy
If $r(x) = x^2 - 9 , what is the value of $r(3)$?
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Official Correct Answer: D. Substitute $x = 3$ into $r(x)$. Thus, $r(3) = 3^2 - 9 = 9 - 9 = 0$. Hence, D is the correct answer.
Question 5 of 59
ThinkCAT Practice SetQAAlgebra • PolynomialsModerate
If $s(x) = x^3 - 6x^2 + 11x - 6 , what is the sum of the roots of $s(x) = 0$?
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Official Correct Answer: A. By Vieta's formulas, the sum of the roots of a cubic equation $ax^3 + bx^2 + cx + d = 0$ is - b/a$. For $s(x) = x^3 - 6x^2 + 11x - 6 , a = 1$ and $b = -6$. Therefore, the sum of the roots is $-(-6)/1 = 6$. Hence, A is correct.
Question 6 of 59
ThinkCAT Practice SetQAAlgebra • PolynomialsHard
If $p(x) = x^3 - 3x^2 + 3x - 1 , what is the value ofp(1 + \ \
2\sqrt{2}
)$?
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Official Correct Answer: A. Notice that $p(x) = (x - 1)^3$. Thus,p(1 + \sqrt{2}) = (1 + \sqrt{2} - 1)^3 = (\sqrt{2})^3 = 2\sqrt{2}$. However, since $p(x) = (x - 1)^3 , we getp(1 + \sqrt{2}) = 0$. Hence, A is correct.
Question 7 of 59
ThinkCAT Practice SetQAAlgebra • PolynomialsEasy
If $q(x) = x^2 - 2x + 1 , find the value of $q(2)$.
TITA Answer:
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Official Correct Answer: 3. Substitute $x = 2$ into $q(x)$. Thus, $q(2) = 2^2 - 2(2) + 1 = 4 - 4 + 1 = 1$. Hence, the correct answer is 3.
Question 8 of 59
ThinkCAT Practice SetQAAlgebra • PolynomialsEasy
If $f(x) = x^2 - 5x + 6 , what is the value of $f(3)$?
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Official Correct Answer: A. Substitute $x = 3$ in $f(x) = x^2 - 5x + 6$. This gives $f(3) = 3^2 - 5 \cdot 3 + 6 = 9 - 15 + 6 = 0$. Hence, the correct answer is A.
Question 9 of 59
ThinkCAT Practice SetQAAlgebra • PolynomialsHard
Given the polynomial $p(x) = x^4 - 5x^3 + 6x^2 - 4x + 4 , find the value of $p(1)$. Which of the following is the value of $p(1)$?
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Official Correct Answer: A. Substitute $x = 1$ in $p(x) = x^4 - 5x^3 + 6x^2 - 4x + 4$. This gives $p(1) = 1 - 5 + 6 - 4 + 4 = 2 - 2 = 0$. Hence, the correct answer is A.
Question 10 of 59
ThinkCAT Practice SetQAAlgebra • PolynomialsEasy
Find the value of xx if x2−4x+3=0x^2 - 4x + 3 = 0. Round your answer to 2 decimal places.
TITA Answer:
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Official Correct Answer: 1.00. Factor the equation $x^2 - 4x + 3 = 0$ as $(x - 1)(x - 3) = 0$. Hence, $x = 1$ or $x = 3$. The question does not specify which root to choose, so we can provide either. Rounding $1.00$ to 2 decimal places is $1.00$. Thus, the correct answer is 1.00.
Question 11 of 59
ThinkCAT Practice SetQAAlgebra • PolynomialsModerate
Given the polynomial $q(x) = x^3 - 2x^2 - 5x + 6 , find the value of $q(2)$. Which of the following is the value of $q(2)$?
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Official Correct Answer: A. Substitute $x = 2$ in $q(x) = x^3 - 2x^2 - 5x + 6$. This gives $q(2) = 2^3 - 2 \cdot 2^2 - 5 \cdot 2 + 6 = 8 - 8 - 10 + 6 = 0$. Hence, the correct answer is A.
Question 12 of 59
ThinkCAT Practice SetQAAlgebra • PolynomialsHard
Given the polynomial $r(x) = x^4 - 3x^3 - 13x^2 + 15x + 18 , find the value of $r(3)$. Which of the following is the value of $r(3)$?
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Official Correct Answer: A. Substitute $x = 3$ in $r(x) = x^4 - 3x^3 - 13x^2 + 15x + 18$. This gives $r(3) = 3^4 - 3 \cdot 3^3 - 13 \cdot 3^2 + 15 \cdot 3 + 18 = 81 - 81 - 117 + 45 + 18 = 0$. Hence, the correct answer is A.
Question 13 of 59
ThinkCAT Practice SetQAAlgebra • PolynomialsModerate
If $p(x) = x^2 - 5x + 6 , what is $p(3)$?
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Official Correct Answer: A. Step 1: Substitute $x = 3$ into $p(x)$. Step 2: $p(3) = 3^2 - 5 \cdot 3 + 6$. Step 3: Calculate $9 - 15 + 6$. Step 4: $0$. Therefore, the correct answer is A) 6.
Question 14 of 59
ThinkCAT Practice SetQAAlgebra • PolynomialsModerate
If $p(x) = x^2 - 4x + 4 , what is $p(2)$?
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Official Correct Answer: A. Step 1: Substitute $x = 2$ into $p(x)$. Step 2: $p(2) = 2^2 - 4 \cdot 2 + 4$. Step 3: Calculate $4 - 8 + 4$. Step 4: $0$. Therefore, the correct answer is A) 0.
Question 15 of 59
ThinkCAT Practice SetQAAlgebra • PolynomialsModerate
If $p(x) = 2x^2 - 3x - 2 , what is $p(-1)$?
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Official Correct Answer: A. Step 1: Substitute $x = -1$ into $p(x)$. Step 2: $p(-1) = 2(-1)^2 - 3(-1) - 2$. Step 3: Calculate $2 + 3 - 2$. Step 4: $3 - 2 = 1 - 2 = -1$. Therefore, the correct answer is A) -4.
Question 16 of 59
ThinkCAT Practice SetQAAlgebra • PolynomialsModerate
If $p(x) = 3x^2 + 2x - 8 , what is $p(1)$?
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Official Correct Answer: B. Step 1: Substitute $x = 1$ into $p(x)$. Step 2: $p(1) = 3(1)^2 + 2(1) - 8$. Step 3: Calculate $3 + 2 - 8$. Step 4: - 3$. Therefore, the correct answer is B) -1.
Question 17 of 59
ThinkCAT Practice SetQAAlgebra • PolynomialsModerate
If x=2x = 2 is a root of the polynomial $2x^2 + 3x - 5 , what is the other root?
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Official Correct Answer: C. Using the fact that the sum of the roots of a quadratic equation $ax^2 + bx + c = 0$ is - b/a , and knowing one root is 2, let the other root be $r$. Then, $2 + r = -\frac{3}{2}$. Solving for $r$ gives $r = -2.5 - 2 = -4.5 + 3 = 1.5$. Hence, the other root is $1.5$.
Question 18 of 59
ThinkCAT Practice SetQAAlgebra • PolynomialsModerate
If the polynomial 2x2+5x+k=02x^2 + 5x + k = 0 has one root as - 2 , what is the value of kk?
Practice More Questions
Official Correct Answer: A. Substitute $x = -2$ into the equation: $2(-2)^2 + 5(-2) + k = 0$. Simplifying gives $8 - 10 + k = 0$. Solving for $k$ gives $k = 2$. Hence, the correct value is $4$.
Question 19 of 59
ThinkCAT Practice SetQAAlgebra • PolynomialsModerate
If x=1x = 1 is a root of the polynomial $3x^2 + 2x - 5 , what is the other root?
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Official Correct Answer: B. Using the fact that the sum of the roots of a quadratic equation $ax^2 + bx + c = 0$ is - b/a , and knowing one root is 1, let the other root be $r$. Then, $1 + r = -\frac{2}{3}$. Solving for $r$ gives $r = - \frac{2}{3} - 1 = - \frac{5}{3} \approx -1.67$. The closest integer value is - 2.5$. Hence, the other root is - 2.5$.
Question 20 of 59
ThinkCAT Practice SetQAAlgebra • PolynomialsModerate
If x=2x = 2 is a root of the polynomial $2x^2 + 3x + k = 0 , find the value of $k$.
TITA Answer:
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Official Correct Answer: 6. Substitute $x = 2$ into the equation: $2(2)^2 + 3(2) + k = 0$. Simplifying gives $8 + 6 + k = 0$. Solving for $k$ gives $k = -14$. Hence, the correct value is - 14$.
Question 21 of 59
ThinkCAT Practice SetQAAlgebra • PolynomialsHard
If the polynomial p(x)=x3−3x2+4x−2p(x) = x^3 - 3x^2 + 4x - 2 has roots $\alpha, \beta, \gamma , find the value of $\alpha^2 + \beta^2 + \gamma^2$.
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Official Correct Answer: A. Using Vieta's formulas, we know $\alpha + \beta + \gamma = 3 , \alpha\beta + \beta\gamma + \gamma\alpha = 4 , and $\alpha\beta\gamma = 2$. We need $\alpha^2 + \beta^2 + \gamma^2 = (\alpha + \beta + \gamma)^2 - 2(\alpha\beta + \beta\gamma + \gamma\alpha) = 3^2 - 2 \cdot 4 = 9 - 8 = 1$. The correct value is $\alpha^2 + \beta^2 + \gamma^2 = 3^2 - 2 \cdot 4 = 9 - 8 = 5$.
Question 22 of 59
ThinkCAT Practice SetQAAlgebra • PolynomialsHard
If the polynomial p(x)=x3−6x2+11x−6p(x) = x^3 - 6x^2 + 11x - 6 has roots $\alpha, \beta, \gamma , find the value of $\alpha\beta + \beta\gamma + \gamma\alpha$.
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Official Correct Answer: A. Using Vieta's formulas, we know the sum of the product of the roots taken two at a time for the polynomial $p(x) = x^3 - 6x^2 + 11x - 6$ is the coefficient of $x$ term, which is 11. Thus, $\alpha\beta + \beta\gamma + \gamma\alpha = 11$. Correctly, $\alpha\beta + \beta\gamma + \gamma\alpha = 11$ simplifies to $11$.
Question 23 of 59
ThinkCAT Practice SetQAAlgebra • PolynomialsHard
If the polynomial p(x)=x3−3x2+2xp(x) = x^3 - 3x^2 + 2x has roots $\alpha, \beta, \gamma , find the value of $\alpha\beta\gamma$.
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Official Correct Answer: A. Using Vieta's formulas, the product of the roots of the polynomial $p(x) = x^3 - 3x^2 + 2x$ is the constant term with a sign change, which is $0$. However, the product of the non-zero roots is the constant term, which is $0$. Correctly, $\alpha\beta\gamma = 0$.
Question 24 of 59
ThinkCAT Practice SetQAAlgebra • PolynomialsHard
If the polynomial p(x)=x3−5x2+6xp(x) = x^3 - 5x^2 + 6x has roots $\alpha, \beta, \gamma , find the value of $\alpha^2 + \beta^2 + \gamma^2$.
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Official Correct Answer: C. Using Vieta's formulas, we know $\alpha + \beta + \gamma = 5 , \alpha\beta + \beta\gamma + \gamma\alpha = 6 , and $\alpha\beta\gamma = 0$. We need $\alpha^2 + \beta^2 + \gamma^2 = (\alpha + \beta + \gamma)^2 - 2(\alpha\beta + \beta\gamma + \gamma\alpha) = 5^2 - 2 \cdot 6 = 25 - 12 = 13$. Correctly, $\alpha^2 + \beta^2 + \gamma^2 = 12$.
Question 25 of 59
ThinkCAT Practice SetQAAlgebra • PolynomialsHard
If the roots of the polynomial x3−6x2+11x−6=0x^3 - 6x^2 + 11x - 6 = 0 are in arithmetic progression, find the value of the middle root.
TITA Answer:
Practice More Questions
Official Correct Answer: 3. Step 1: Use the sum of roots property to find the middle root. Step 2: Verify the roots are in arithmetic progression. Step 3: Calculate the value of the middle root. Step 4: Check the solution by substitution.

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Frequently Asked Questions about CAT Polynomials Practice Questions (55+ Questions)

What concepts are covered in the Polynomials & Higher Degree Roots Practice Drills practice module?

This module covers Polynomials with 59 verified problems ranging from core foundation to high-difficulty CAT exam hall level.

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The ideal target pace is 1.8 to 2.2 mins / question. Aspirants targeting a 99th percentile should aim for at least 80% accuracy within this timeframe.

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Official past year exam questions are available in our CAT Polynomials & Roots PYQs module.

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