Click on a question to see the answer and explanation.
▶1. What is the degree of the polynomial P(x) = 4x³ - 2x² + 7?
Answer: 3
Explanation: The degree of a polynomial is the highest power of x. Here, the highest power is 3 (from 4x³). So, the degree is 3.
▶2. Identify the constant term in the polynomial Q(x) = 5x² - 3x + 8.
Answer: 8
Explanation: The constant term is the term without any variable. Here, it is 8.
▶3. Is 7x⁴ - 2x + 5 a quadratic polynomial?
Answer: No
Explanation: A quadratic polynomial has degree 2. Here, the highest degree is 4, so it is not quadratic.
▶4. Add the polynomials: (2x² + 3x + 1) and (x² - x + 4).
Answer: 3x² + 2x + 5
Explanation: Add like terms:
- 2x² + x² = 3x²
- 3x + (-x) = 2x
- 1 + 4 = 5
▶5. Subtract (x² + 2x - 3) from (3x² - x + 5).
Answer: 2x² - 3x + 8
Explanation: (3x² - x + 5) - (x² + 2x - 3) = (3x² - x²) + (-x - 2x) + (5 - (-3)) = 2x² - 3x + 8
▶6. Multiply: (x + 2)(x - 5)
Answer: x² - 3x - 10
Explanation: (x + 2)(x - 5) = x(x - 5) + 2(x - 5) = x² - 5x + 2x - 10 = x² - 3x - 10
▶7. Factor: x² - 9
Answer: (x + 3)(x - 3)
Explanation: This is a difference of squares: x² - 9 = (x + 3)(x - 3)
▶8. Find the value of the polynomial P(x) = 2x² - 3x + 4 at x = 2.
Answer: 6
Explanation: Substitute x = 2: 2(2)² - 3(2) + 4 = 2(4) - 6 + 4 = 8 - 6 + 4 = 6
▶9. What is the sum of the roots of x² - 7x + 10 = 0?
Answer: 7
Explanation: For ax² + bx + c = 0, sum of roots = -b/a. Here, a = 1, b = -7, so sum = -(-7)/1 = 7
▶10. If x = 1 is a root of P(x) = x² - 2x + k, find k.
Answer: 1
Explanation: Substitute x = 1: 1² - 2(1) + k = 0 ⇒ 1 - 2 + k = 0 ⇒ k = 1
▶11. Divide x³ - 8 by x - 2 and state the remainder.
Answer: 0
Explanation: By Remainder Theorem, remainder = P(2) = (2)³ - 8 = 8 - 8 = 0
▶12. Is x² + 2x + 1 a perfect square trinomial?
Answer: Yes
Explanation: x² + 2x + 1 = (x + 1)², which is a perfect square trinomial.
▶13. Find the product of the roots of x² - 5x + 6 = 0.
Answer: 6
Explanation: For ax² + bx + c = 0, product of roots = c/a. Here, c = 6, a = 1, so product = 6/1 = 6
▶14. Expand (x + 3)².
Answer: x² + 6x + 9
Explanation: (x + 3)² = x² + 2×3x + 3² = x² + 6x + 9
▶15. If P(x) = x² - 4x + 3, find P(0).
Answer: 3
Explanation: Substitute x = 0: (0)² - 4×0 + 3 = 0 - 0 + 3 = 3
▶16. Factor: x² + 5x + 6
Answer: (x + 2)(x + 3)
Explanation: Find two numbers that multiply to 6 and add to 5: 2 and 3. So, x² + 5x + 6 = (x + 2)(x + 3)
▶17. What is the coefficient of x in 7x² - 4x + 9?
Answer: -4
Explanation: The coefficient of x is the number in front of x, which is -4.
▶18. If x = -1 is a root of x² + kx + 2 = 0, find k.
Answer: -3
Explanation: Substitute x = -1: (-1)² + k(-1) + 2 = 0 ⇒ 1 - k + 2 = 0 ⇒ k = 3
▶19. Expand (x - 2)(x + 5).
Answer: x² + 3x - 10
Explanation: (x - 2)(x + 5) = x² + 5x - 2x - 10 = x² + 3x - 10
▶20. If P(x) = x³ - 6x² + 11x - 6, find the sum of its roots.
Answer: 6
Explanation: For a cubic ax³ + bx² + cx + d, sum of roots = -b/a. Here, b = -6, a = 1, so sum = -(-6)/1 = 6