Polynomials — Important Questions

1/x + 3

Step 1: A polynomial must have non-negative integer powers of x. Step 2: Check each option: 3x² + 2x - 1 has powers 2, 1, 0 (all valid). Step 3: 1/x + 3 = x⁻¹ + 3 has power -1 (negative power). Step 4: 5x³ - x + 7 and 2x - 9 have valid powers. Therefore, 1/x + 3 is NOT a polynomial.

3 and 4

Step 1: Factor x² - 7x + 12. We need two numbers that multiply to 12 and add to -7. Step 2: The numbers are -3 and -4 since (-3) × (-4) = 12 and (-3) + (-4) = -7. Step 3: x² - 7x + 12 = (x - 3)(x - 4). Step 4: Set each factor to zero: x - 3 = 0 gives x = 3, and x - 4 = 0 gives x = 4. Therefore, the zeroes are 3 and 4.

-3/2

Step 1: For a quadratic polynomial ax² + bx + c, the sum of zeroes = -b/a. Step 2: Compare 2x² + 3x - 5 with ax² + bx + c: a = 2, b = 3, c = -5. Step 3: Sum of zeroes = -b/a = -3/2. Step 4: Therefore, the sum of the zeroes is -3/2.

3

Step 1: For a quadratic polynomial ax² + bx + c, the product of zeroes = c/a. Step 2: Compare x² - 4x + 3 with ax² + bx + c: a = 1, b = -4, c = 3. Step 3: Product of zeroes = c/a = 3/1 = 3. Step 4: Therefore, the product of the zeroes is 3.