Question 1

What is the remainder when p(x) = x³ + 2x² - 5x + 3 is divided by (x - 1)?

Accepted Answer

Step 1: Use the Remainder Theorem: when p(x) is divided by (x - a), the remainder is p(a). Step 2: Here, we divide by (x - 1), so a = 1. Step 3: Calculate p(1) = (1)³ + 2(1)² - 5(1) + 3 = 1 + 2 - 5 + 3 = 1.

Question 2

What is the degree of the polynomial 4x³ + 2x² - 7x + 5?

Accepted Answer

Step 1: Identify the highest power of x in the polynomial. Step 2: The term 4x³ has the highest power, which is 3. Step 3: Therefore, the degree is 3.

Question 3

If (x + 3) is a factor of the polynomial x² + 7x + 12, what is the other linear factor?

Accepted Answer

Step 1: Factor the polynomial x² + 7x + 12. Step 2: Find two numbers that multiply to 12 and add to 7: these are 3 and 4. Step 3: x² + 7x + 12 = (x + 3)(x + 4). Step 4: Since (x + 3) is given as one factor, the other factor is (x + 4).

Question 4

Simplify: (2x + 3)(2x - 3)

Accepted Answer

Step 1: Recognize this as the difference of squares pattern: (a + b)(a - b) = a² - b². Step 2: Here, a = 2x and b = 3. Step 3: (2x + 3)(2x - 3) = (2x)² - 3² = 4x² - 9.

Question 5

If the polynomial p(x) = x² - 5x + 6 has roots α and β, what is the value of α + β?

Accepted Answer

Step 1: For a quadratic polynomial ax² + bx + c, the sum of roots = -b/a. Step 2: In p(x) = x² - 5x + 6, we have a = 1, b = -5, c = 6. Step 3: Sum of roots = -(-5)/1 = 5.

Question 6

If (x - 3) is a factor of the polynomial q(x) = x³ - 6x² + 11x - 6, what is the remainder when q(x) is divided by (x - 3)?

Accepted Answer

Step 1: By the Factor Theorem, if (x - 3) is a factor of q(x), then q(3) = 0. Step 2: Calculate q(3) = (3)³ - 6(3)² + 11(3) - 6 = 27 - 54 + 33 - 6 = 0. Step 3: Since q(3) = 0, the remainder is 0, confirming (x - 3) is indeed a factor.

Question 7

If the polynomial r(x) = x² + bx + 8 has roots α and β such that α + β = -5, find the value of b.

Accepted Answer

Step 1: For a quadratic polynomial x² + bx + c, the sum of roots is given by α + β = -b (Vieta's formula). Step 2: Given that α + β = -5, we have -b = -5. Step 3: Therefore, b = 5.

Question 8

If the polynomial t(x) = 2x³ - 3x² + px - 10 leaves a remainder of 4 when divided by (x - 2), find the value of p.

Accepted Answer

Step 1: By the Remainder Theorem, when t(x) is divided by (x - 2), the remainder is t(2). Step 2: Given that the remainder is 4, we have t(2) = 4. Step 3: Substitute x = 2: t(2) = 2(2)³ - 3(2)² + p(2) - 10 = 16 - 12 + 2p - 10 = 2p - 6. Step 4: Set 2p - 6 = 4, so 2p = 10, thus p = 5.

Question 9

The polynomial u(x) = x⁴ - 5x³ + 5x² + 5x - 6 can be factored. If (x - 1) and (x + 1) are both factors, what is the product of the other two roots?

Accepted Answer

Step 1: If (x - 1) and (x + 1) are factors, then the roots include 1 and -1. Step 2: For a monic polynomial of degree 4, if the roots are r₁, r₂, r₃, r₄, then by Vieta's formula, the product of all roots equals the constant term (with appropriate sign): r₁ × r₂ × r₃ × r₄ = (-1)⁴ × (-6) = -6. Step 3: The product of the known roots is 1 × (-1) = -1. Step 4: If the other two roots are α and β, then (-1) × α × β = -6, so α × β = 6.

Question 10

If the polynomial x⁴ + px³ + qx² + rx + s has roots in arithmetic progression with common difference 2, and the sum of roots is 4, find the value of p.

Accepted Answer

Step 1: Let the roots be (a - 3), (a - 1), (a + 1), (a + 3) in AP with common difference 2. Step 2: Sum of roots = (a - 3) + (a - 1) + (a + 1) + (a + 3) = 4a = 4, so a = 1. Step 3: The roots are -2, 0, 2, 4. Step 4: By Vieta's formulas, for a monic polynomial x⁴ + px³ + ..., the coefficient p equals the negative of the sum of roots. Step 5: p = -(−2 + 0 + 2 + 4) = −4.

Question 11

If p(x) = x³ - 6x² + 11x - 6 and p(x) = (x - a)(x - b)(x - c), find the value of (1/a) + (1/b) + (1/c).

Accepted Answer

Step 1: Expand (x - a)(x - b)(x - c) = x³ - (a+b+c)x² + (ab+bc+ca)x - abc. Step 2: Compare with p(x) = x³ - 6x² + 11x - 6: a + b + c = 6, ab + bc + ca = 11, abc = 6. Step 3: Calculate (1/a) + (1/b) + (1/c) = (bc + ac + ab)/(abc) = (ab + bc + ca)/(abc) = 11/6.

Question 12

If p(x) = x³ - 6x² + 11x - 6 and q(x) = x² - 3x + 2, find the remainder when p(x) is divided by q(x).

Accepted Answer

Step 1: Use polynomial long division or the remainder theorem. Since q(x) is quadratic, the remainder must be linear: R(x) = ax + b. Step 2: Write p(x) = q(x)·d(x) + R(x). Step 3: Divide x³ - 6x² + 11x - 6 by x² - 3x + 2. First term of quotient: x³ ÷ x² = x. Multiply: x(x² - 3x + 2) = x³ - 3x² + 2x. Subtract: (x³ - 6x² + 11x - 6) - (x³ - 3x² + 2x) = -3x² + 9x - 6. Step 4: Next term: -3x² ÷ x² = -3. Multiply: -3(x² - 3x + 2) = -3x² + 9x - 6. Subtract: (-3x² + 9x - 6) - (-3x² + 9x - 6) = 0. Step 5: Remainder is 0.

Question 13

If α and β are roots of x² - 5x + 6 = 0, and γ and δ are roots of x² - 7x + 12 = 0, find the value of (α + β + γ + δ)² - 4(αβ + γδ).

Accepted Answer

Step 1: For x² - 5x + 6 = 0, by Vieta's formulas: α + β = 5 and αβ = 6. Step 2: For x² - 7x + 12 = 0, by Vieta's formulas: γ + δ = 7 and γδ = 12. Step 3: Calculate α + β + γ + δ = 5 + 7 = 12. Step 4: Calculate αβ + γδ = 6 + 12 = 18. Step 5: Substitute into expression: (12)² - 4(18) = 144 - 72 = 72.

IBPS Clerk Polynomials

IBPS Clerk Polynomials — Past Exam Questions

Polynomials— Rules & Concept

Key Points to Remember

Exam-Specific Tips

60-Second Revision — Polynomials