Question 1

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)² - 3² = 4x² - 9.

Question 2

What is the coefficient of x in the expansion of (x + 2)³?

Accepted Answer

Step 1: Expand (x + 2)³ using the binomial theorem or direct multiplication. Step 2: (x + 2)³ = x³ + 3(x²)(2) + 3(x)(2²) + 2³ = x³ + 6x² + 12x + 8. Step 3: The coefficient of x is 12.

Question 3

If p(x) = x³ - 4x² + 5x - 2, find p(1).

Accepted Answer

Step 1: Substitute x = 1 into p(x) = x³ - 4x² + 5x - 2. Step 2: p(1) = (1)³ - 4(1)² + 5(1) - 2 = 1 - 4 + 5 - 2 = 0.

Question 4

What is the degree of the polynomial 4x⁵ + 3x³ - 2x + 7?

Accepted Answer

Step 1: Identify the highest power of x in the polynomial. Step 2: The terms are 4x⁵, 3x³, -2x, and 7. Step 3: The highest exponent is 5, so the degree is 5.

Question 5

If (x + 3) is a factor of the polynomial p(x) = x² + 5x + 6, what is the other factor?

Accepted Answer

Step 1: Factor the polynomial x² + 5x + 6. Step 2: Find two numbers that multiply to 6 and add to 5: these are 2 and 3. Step 3: x² + 5x + 6 = (x + 2)(x + 3). Step 4: Since (x + 3) is one factor, the other factor is (x + 2).

Question 6

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

Accepted Answer

Step 1: By Vieta's formulas, for a quadratic x² + bx + c, the sum of roots α + β = -b. Step 2: Given α + β = -5, we have -b = -5. Step 3: Therefore, b = 5.

Question 7

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: 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.

Question 8

If the polynomial s(x) = x³ + ax² - 7x + 6 is divisible by (x + 2), find the value of a.

Accepted Answer

Step 1: If s(x) is divisible by (x + 2), then s(-2) = 0 (by the Factor Theorem). Step 2: s(-2) = (-2)³ + a(-2)² - 7(-2) + 6 = -8 + 4a + 14 + 6 = 0. Step 3: 4a + 12 = 0, so 4a = -12, thus a = -3.

Question 9

The polynomial t(x) = x³ - 3x² - 10x + 24 can be factored as (x - 2)(x² + bx + c). Find b + c.

Accepted Answer

Step 1: If t(x) = (x - 2)(x² + bx + c), expand: (x - 2)(x² + bx + c) = x³ + bx² + cx - 2x² - 2bx - 2c = x³ + (b - 2)x² + (c - 2b)x - 2c. Step 2: Compare with t(x) = x³ - 3x² - 10x + 24: b - 2 = -3 → b = -1; c - 2b = -10 → c - 2(-1) = -10 → c + 2 = -10 → c = -12; verify: -2c = 24 → c = -12 ✓. Step 3: b + c = -1 + (-12) = -13.

Question 10

If the polynomial u(x) = 2x³ + 3x² - 8x - 12 has a factor (x + 2), find the other quadratic factor in the form 2x² + px + q. What is p?

Accepted Answer

Step 1: Since (x + 2) is a factor, divide u(x) by (x + 2) using synthetic division or long division. Step 2: u(x) ÷ (x + 2): 2x³ + 3x² - 8x - 12 = (x + 2)(2x² - x - 6). Step 3: Verify: (x + 2)(2x² - x - 6) = 2x³ - x² - 6x + 4x² - 2x - 12 = 2x³ + 3x² - 8x - 12 ✓. Step 4: The quadratic factor is 2x² - x - 6, so p = -1.

Question 11

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), where d(x) is the quotient. Step 3: Perform division: x³ - 6x² + 11x - 6 = (x² - 3x + 2)(x - 3) + 0. Step 4: The remainder is 0.

Question 12

If α and β are roots of x² - 5x + 6 = 0, and γ and δ are roots of x² - 7x + 12 = 0, then find the sum of all possible products αγ + αδ + βγ + βδ.

Accepted Answer

Step 1: Find roots of first equation: x² - 5x + 6 = (x-2)(x-3), so α = 2, β = 3. Step 2: Find roots of second equation: x² - 7x + 12 = (x-3)(x-4), so γ = 3, δ = 4. Step 3: Calculate αγ + αδ + βγ + βδ = (2)(3) + (2)(4) + (3)(3) + (3)(4) = 6 + 8 + 9 + 12 = 35.

Question 13

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 product of the roots.

Accepted Answer

Step 1: Let 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: Roots are -2, 0, 2, 4. Step 4: By Vieta's formulas, product of roots = s/1 = (-2)(0)(2)(4) = 0.

Question 14

If p(x) = x⁴ - 10x³ + 35x² - 50x + 24, and it is known that p(x) = (x-1)(x-2)(x-3)(x-4), find the remainder when p(x) is divided by (x² - 5x + 6).

Accepted Answer

Step 1: Note that x² - 5x + 6 = (x-2)(x-3). Step 2: Since p(x) = (x-1)(x-2)(x-3)(x-4), we have p(x) = (x-2)(x-3)·(x-1)(x-4). Step 3: When p(x) is divided by (x-2)(x-3), the quotient is (x-1)(x-4) = x² - 5x + 4 and the remainder is 0. Step 4: Therefore, remainder = 0.

Question 15

A cubic polynomial p(x) = x³ + ax² + bx + c has the property that p(1) = 2, p(2) = 9, and p(3) = 28. Find the value of a + b + c.

Accepted Answer

Step 1: We have three equations: 1 + a + b + c = 2, 8 + 4a + 2b + c = 9, 27 + 9a + 3b + c = 28. Step 2: Simplify: a + b + c = 1, 4a + 2b + c = 1, 9a + 3b + c = 1. Step 3: From equation 1: a + b + c = 1. Step 4: From equation 2 - equation 1: 3a + b = 0, so b = -3a. Step 5: From equation 3 - equation 2: 5a + b = 0, so b = -5a. Step 6: From b = -3a and b = -5a: -3a = -5a, so 2a = 0, thus a = 0. Step 7: Then b = 0 and c = 1. Step 8: Therefore a + b + c = 1.

SBI Clerk Polynomials

SBI Clerk Polynomials — Past Exam Questions

Polynomials— Rules & Concept

Key Points to Remember

Exam-Specific Tips

60-Second Revision — Polynomials