Polynomials — Practice

Question 1

Find the zeros of 4x² − 4x − 3 and verify the sum/product of zeros.

Accepted Answer

Zeros: 3/2 and −1/2. Sum = 1 = −(−4)/4 ✓. Product = −3/4 = c/a = −3/4 ✓.

Question 2

Find a quadratic polynomial with zeros 2 and −1/3.

Accepted Answer

Sum = 5/3, product = −2/3. Polynomial = k·(x² − (5/3)x − 2/3). Multiply by 3: 3x² − 5x − 2.

Question 3

Divide p(x) = x³ − 3x² + 3x − 5 by g(x) = x − 1.

Accepted Answer

Long division gives quotient x² − 2x + 1 and remainder −4. Check: (x−1)(x²−2x+1) + (−4) = x³ − 3x² + 3x − 5 ✓.

Question 4

If α, β are zeros of x² − p(x+1) − c, find (α+1)(β+1).

Accepted Answer

Expand: x² − px − (p + c) so α + β = p and αβ = −(p+c). Then (α+1)(β+1) = αβ + α + β + 1 = −(p+c) + p + 1 = 1 − c.

Practice — Polynomials