Polynomials

x = 2, x = 3

To find zeros, set p(x) = 0: Step 1: x² - 5x + 6 = 0 Step 2: Factor: (x - 2)(x - 3) = 0 Step 3: Therefore x = 2 or x = 3 Verification: p(2) = 4 - 10 + 6 = 0 ✓, p(3) = 9 - 15 + 6 = 0 ✓

-1

Step 1: Sum of zeros = 3 + (-2) = 1 Step 2: For x² + px + q, sum of zeros = -p/1 = -p Step 3: Therefore, -p = 1, so p = -1 Verification: If p = -1, the polynomial is x² - x + q, and its zeros should sum to 1, which matches 3 + (-2) = 1 ✓

2x² + 3x - 1, -x² + 4, x² - 2x + 3x²

A quadratic polynomial has degree 2 (highest power of x is 2): • 2x² + 3x - 1: degree 2 ✓ • x³ + 2x² - 5: degree 3 ✗ • 5x + 7: degree 1 ✗ • -x² + 4: degree 2 ✓ • x² - 2x + 3x² = 4x² - 2x: degree 2 ✓

1

If x = -1 is a zero, then p(-1) = 0 Step 1: Substitute x = -1 in 2x² + 3x + k Step 2: 2(-1)² + 3(-1) + k = 0 Step 3: 2(1) - 3 + k = 0 Step 4: 2 - 3 + k = 0 Step 5: -1 + k = 0 Step 6: k = 1