Polynomials: Degree, Zeroes, Remainder and Factor Theorems, and Zero-Coefficient Relations

Polynomials: Remainders Without Long Division

A polynomial is an expression such as 3x² − 5x + 2, built from a variable raised to whole-number powers and multiplied by constants. Expressions such as √x or 1/x are not polynomials, because their powers are not whole numbers.

Exam algebra keeps returning to a few polynomial ideas: the degree, the zeroes, and above all the remainder when one polynomial is divided by another. Long division answers that slowly. The remainder theorem and the factor theorem answer it in one substitution, often in under twenty seconds.

This short builds on, and does not repeat, the separate topic on quadratic equations and roots. Here the focus is the two theorems, finding unknown constants, and the zero-coefficient relations for quadratics and cubics. Every card has worked numbers you can check by hand.

1. What is the degree of the polynomial 5x⁴ − 3x⁶ + 2x + 7?

2. What is the remainder when x³ − 3x² + 4x − 5 is divided by (x − 2)?

3. If (x + 2) is a factor of x³ + kx² − x + 2, what is the value of k?

4. For the quadratic polynomial 2x² − 5x + 3, the sum and product of the zeroes are:

5. Which of the following is a factor of x³ − 6x² + 11x − 6?

6. What is the product of the zeroes of the cubic polynomial 2x³ − 3x² + 4x − 8?

7. What is the remainder when x⁴ + 1 is divided by (x + 1)?

8. Which quadratic polynomial has zeroes 2 and −3?