Question 1

Simplify: (x + 3)(x - 2) - (x + 1)²

Accepted Answer

Step 1: Expand (x + 3)(x - 2) = x² - 2x + 3x - 6 = x² + x - 6. Step 2: Expand (x + 1)² = x² + 2x + 1. Step 3: Subtract: (x² + x - 6) - (x² + 2x + 1) = x² + x - 6 - x² - 2x - 1 = -x - 7.

Question 2

Find the sum of coefficients of the polynomial h(x) = 2x³ + 5x² - 3x + 4.

Accepted Answer

Step 1: The sum of coefficients of any polynomial is found by substituting x = 1. Step 2: h(1) = 2(1)³ + 5(1)² - 3(1) + 4 = 2 + 5 - 3 + 4 = 8.

Question 3

If q(x) = x² - 6x + 8, find the sum of the roots of q(x) = 0.

Accepted Answer

Step 1: For a quadratic polynomial ax² + bx + c, the sum of roots = -b/a. Step 2: In q(x) = x² - 6x + 8, we have a = 1, b = -6, c = 8. Step 3: Sum of roots = -(-6)/1 = 6.

Question 4

Find the product of the roots of the polynomial r(x) = 2x² + 3x - 5.

Accepted Answer

Step 1: For a quadratic polynomial ax² + bx + c, the product of roots = c/a. Step 2: In r(x) = 2x² + 3x - 5, we have a = 2, c = -5. Step 3: Product of roots = -5/2 = -2.5.

Question 5

What is the remainder when the polynomial t(x) = x² + 2x + 1 is divided by (x + 1)?

Accepted Answer

Step 1: By the Remainder Theorem, when a polynomial p(x) is divided by (x - a), the remainder is p(a). Step 2: Here, we divide by (x + 1) = (x - (-1)), so a = -1. Step 3: Remainder = t(-1) = (-1)² + 2(-1) + 1 = 1 - 2 + 1 = 0.

Question 6

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

Accepted Answer

Step 1: If (x + 3) is a factor of p(x), then p(-3) must equal 0 (by the Factor Theorem). Step 2: Verify by substitution: p(-3) = (-3)² + 5(-3) + 6 = 9 - 15 + 6 = 0. Therefore, p(-3) = 0.

Question 7

Simplify: (x + 4)(x - 2) - (x + 1)²

Accepted Answer

Step 1: Expand (x + 4)(x - 2) = x² - 2x + 4x - 8 = x² + 2x - 8. Step 2: Expand (x + 1)² = x² + 2x + 1. Step 3: Subtract: (x² + 2x - 8) - (x² + 2x + 1) = x² + 2x - 8 - x² - 2x - 1 = -9. Therefore, the answer is -9.

Question 8

If s(x) = x³ - 4x² + x + 6, what is the value of s(-1)?

Accepted Answer

Step 1: Substitute x = -1 into s(x) = x³ - 4x² + x + 6. Step 2: s(-1) = (-1)³ - 4(-1)² + (-1) + 6 = -1 - 4(1) - 1 + 6 = -1 - 4 - 1 + 6 = 0.

Question 9

What is the degree of the polynomial q(x) = 7x⁴ + 3x² - 2x + 8?

Accepted Answer

Step 1: Identify all terms in the polynomial: 7x⁴, 3x², -2x, and 8. Step 2: The degree is the highest power of x, which is 4.

Question 10

If r(x) = x³ - 6x² + 11x - 6, find r(1).

Accepted Answer

Step 1: Substitute x = 1 into r(x) = x³ - 6x² + 11x - 6. Step 2: r(1) = (1)³ - 6(1)² + 11(1) - 6 = 1 - 6 + 11 - 6 = 0.

Question 11

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

Accepted Answer

Step 1: The degree of a polynomial is the highest power of the variable x. Step 2: In 4x⁵ + 3x³ - 2x + 7, the highest power is 5. Step 3: Therefore, the degree is 5.

Question 12

If the polynomial r(x) = x³ + ax² - 7x + b is divisible by (x - 2) and (x + 1), find the value of a + b.

Accepted Answer

Step 1: If r(x) is divisible by (x - 2), then r(2) = 0. Step 2: r(2) = 8 + 4a - 14 + b = 0 ⟹ 4a + b = 6. Step 3: If r(x) is divisible by (x + 1), then r(-1) = 0. Step 4: r(-1) = -1 + a + 7 + b = 0 ⟹ a + b = -6. Step 5: From equation 4a + b = 6 and a + b = -6, subtract: 3a = 12, so a = 4. Step 6: Substitute a = 4 into a + b = -6: 4 + b = -6, so b = -10. Step 7: a + b = 4 + (-10) = -6. Therefore, the answer is -6.

Question 13

The polynomial s(x) = 2x³ - 3x² - 8x + 12 can be factored as 2(x - p)(x - q)(x - r). If p, q, r are the roots, what is p + q + r?

Accepted Answer

Step 1: For a cubic polynomial s(x) = 2x³ - 3x² - 8x + 12, rewrite in standard form: 2x³ - 3x² - 8x + 12 = 2(x³ - (3/2)x² - 4x + 6). Step 2: By Vieta's formulas for a monic cubic x³ + Ax² + Bx + C, the sum of roots = -A. Step 3: Here, the coefficient of x² in the monic form is -3/2, so sum of roots = -(-3/2) = 3/2. Step 4: Therefore, p + q + r = 3/2. Alternatively, for s(x) = 2x³ - 3x² - 8x + 12, sum of roots = (coefficient of x²) / (coefficient of x³) with opposite sign = -(-3)/2 = 3/2. Therefore, the answer is 3/2 or 1.5.

Question 14

If the polynomial s(x) = x³ - 4x² + ax + b is divisible by (x - 1) and (x - 2), find the value of a + b.

Accepted Answer

Step 1: If s(x) is divisible by (x - 1) and (x - 2), then s(1) = 0 and s(2) = 0. Step 2: s(1) = 1 - 4 + a + b = 0 ⟹ a + b = 3. Step 3: s(2) = 8 - 16 + 2a + b = 0 ⟹ 2a + b = 8. Step 4: From equation 2: a + b = 3. Therefore, a + b = 3.

Question 15

If (2x + 3) is a factor of the polynomial t(x) = 2x³ + 5x² - 4x - 3, what is the remainder when t(x) is divided by (x - 1)?

Accepted Answer

Step 1: To find the remainder when t(x) is divided by (x - 1), use the Remainder Theorem: remainder = t(1). Step 2: t(1) = 2(1)³ + 5(1)² - 4(1) - 3 = 2 + 5 - 4 - 3 = 0. Step 3: Therefore, the remainder is 0. Note: The fact that (2x + 3) is a factor is given context but not needed for this calculation; the Remainder Theorem directly gives the answer.

Question 16

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

Accepted Answer

Step 1: Since (x - 3) is a factor, q(3) = 0. Verify: 27 - 54 + 33 - 6 = 0 ✓. Step 2: Perform polynomial division of x³ - 6x² + 11x - 6 by (x - 3). Step 3: x³ ÷ x = x², then x²(x - 3) = x³ - 3x². Subtract: (x³ - 6x² + 11x - 6) - (x³ - 3x²) = -3x² + 11x - 6. Step 4: -3x² ÷ x = -3x, then -3x(x - 3) = -3x² + 9x. Subtract: (-3x² + 11x - 6) - (-3x² + 9x) = 2x - 6. Step 5: 2x ÷ x = 2, then 2(x - 3) = 2x - 6. Subtract: (2x - 6) - (2x - 6) = 0. Therefore, quotient is x² - 3x + 2.

Question 17

The polynomial r(x) = x² - 7x + k has roots α and β. If α + β = 7 and αβ = 10, find the value of k.

Accepted Answer

Step 1: By Vieta's formulas for a quadratic x² + bx + c, the sum of roots = -b and product of roots = c. Step 2: For r(x) = x² - 7x + k, sum of roots = 7 and product of roots = k. Step 3: Given αβ = 10, we have k = 10.

Question 18

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

Accepted Answer

Step 1: Since (x - 3) is a factor, we perform polynomial division or use the fact that q(x) = (x - 3) × quotient. Step 2: Divide x³ - 6x² + 11x - 6 by (x - 3). First term: x³ ÷ x = x². Multiply: x²(x - 3) = x³ - 3x². Subtract: (x³ - 6x² + 11x - 6) - (x³ - 3x²) = -3x² + 11x - 6. Step 3: Second term: -3x² ÷ x = -3x. Multiply: -3x(x - 3) = -3x² + 9x. Subtract: (-3x² + 11x - 6) - (-3x² + 9x) = 2x - 6. Step 4: Third term: 2x ÷ x = 2. Multiply: 2(x - 3) = 2x - 6. Subtract: (2x - 6) - (2x - 6) = 0. Quotient is x² - 3x + 2.

Question 19

The polynomial u(x) = x³ + px² + qx + r has roots 2, 3, and 5. Find the value of p + q + r.

Accepted Answer

Step 1: By Vieta's formulas for a cubic x³ + px² + qx + r with roots α, β, γ: sum of roots = -p, sum of products of pairs = q, product of roots = -r. Step 2: Sum of roots: 2 + 3 + 5 = 10 = -p ⟹ p = -10. Step 3: Sum of products of pairs: (2)(3) + (3)(5) + (5)(2) = 6 + 15 + 10 = 31 = q. Step 4: Product of roots: (2)(3)(5) = 30 = -r ⟹ r = -30. Step 5: p + q + r = -10 + 31 + (-30) = -9.

Question 20

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) - 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 = 35. Step 4: Calculate αβ·γδ = 6 × 12 = 72. Step 5: (α + β)(γ + δ) - αβ·γδ = 35 - 72 = -37.

RRB NTPC Polynomials

Polynomials— Rules & Concept

Key Points to Remember

Exam-Specific Tips

Polynomials — Practice Questions

60-Second Revision — Polynomials