Polynomials — Practice

Question 1

State the degree of the polynomial 5x³ − 4x² + 7x − 2.

Accepted Answer

The highest power of x is 3.

✦ Answer: Degree 3 (a cubic polynomial).

Question 2

Is x = 2 a zero of p(x) = x² − 5x + 6?

Accepted Answer

Step 1 — p(2) = 4 − 10 + 6 = 0.

✦ Answer: Yes, since p(2) = 0, x = 2 is a zero.

Question 3

For x² − 7x + 10, write the sum and product of its zeroes without solving.

Accepted Answer

Step 1 — a = 1, b = −7, c = 10.

Step 2 — Sum = −b/a = 7; Product = c/a = 10.

✦ Answer: sum = 7, product = 10.

Question 4

Find the zeroes of x² − 2x − 8 and verify the relationship with its coefficients.

Accepted Answer

Step 1 — Factorise: x² − 2x − 8 = (x − 4)(x + 2). Zeroes: 4 and −2.

Step 2 — Sum = 4 + (−2) = 2; −b/a = −(−2)/1 = 2. ✓

Step 3 — Product = 4 × (−2) = −8; c/a = −8/1 = −8. ✓

✦ Answer: zeroes 4 and −2; relationships verified.

Question 5

Find a quadratic polynomial whose zeroes have sum 4 and product 1.

Accepted Answer

Step 1 — Quadratic = x² − (sum)x + (product).

Step 2 — = x² − 4x + 1.

✦ Answer: x² − 4x + 1.

Question 6

Find the sum and product of the zeroes of 2x² − 5x + 3.

Accepted Answer

Step 1 — a = 2, b = −5, c = 3.

Step 2 — Sum = −b/a = 5/2.

Step 3 — Product = c/a = 3/2.

✦ Answer: sum = 5/2, product = 3/2.

Question 7

If α and β are the zeroes of x² − 6x + 8, find α² + β².

Accepted Answer

Step 1 — Sum α + β = 6; product αβ = 8.

Step 2 — Use α² + β² = (α + β)² − 2αβ.

Step 3 — = 6² − 2(8) = 36 − 16 = 20.

✦ Answer: α² + β² = 20.

Question 8

Find a quadratic polynomial whose zeroes are 3 + √2 and 3 − √2.

Accepted Answer

Step 1 — Sum = (3 + √2) + (3 − √2) = 6.

Step 2 — Product = (3 + √2)(3 − √2) = 9 − 2 = 7.

Step 3 — Quadratic = x² − (sum)x + product = x² − 6x + 7.

✦ Answer: x² − 6x + 7.

Question 9

If one zero of x² − 4x + k is 3, find k and the other zero.

Accepted Answer

Step 1 — Since 3 is a zero, p(3) = 0: 9 − 12 + k = 0 ⇒ k = 3.

Step 2 — Sum of zeroes = −b/a = 4, so other zero = 4 − 3 = 1.

Step 3 — Check: product = c/a = k = 3 = 3 × 1. ✓

✦ Answer: k = 3, other zero = 1.

Question 10

If α and β are the zeroes of x² − 5x + 6, find a quadratic polynomial whose zeroes are 1/α and 1/β.

Accepted Answer

Step 1 — For x² − 5x + 6: α + β = 5, αβ = 6.

Step 2 — New sum = 1/α + 1/β = (α + β)/(αβ) = 5/6.

Step 3 — New product = 1/(αβ) = 1/6.

Step 4 — Polynomial = x² − (5/6)x + 1/6, or multiplying by 6: 6x² − 5x + 1.

✦ Answer: 6x² − 5x + 1.

Practice — Polynomials

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