Master Polynomial Operations with This Test

What Is a Polynomial? The Basics You Need First

A polynomial is a mathematical expression with variables and coefficients, combined using addition, subtraction, and multiplication. Exponents are always whole numbers—no fractions, no negatives.

Examples:

• 3x² + 2x - 5
• 4a³ - 7a² + a + 9
• 2xy + 3y² - 8

The degree of a polynomial is the highest exponent. 5x³ + 2x² - 3x has degree 3. This matters for division and factoring.

Adding and Subtracting Polynomials

Combine like terms only. "Like terms" means same variables raised to same powers.

Example:

(3x² + 5x - 2) + (4x² - 3x + 7)

Add coefficients of x²: 3 + 4 = 7x²

Add coefficients of x: 5 + (-3) = 2x

Add constants: -2 + 7 = 5

Answer: 7x² + 2x + 5

Subtraction works the same way—just distribute the negative sign first:

(5x³ + 3x²) - (2x³ - 4x²) = 5x³ + 3x² - 2x³ + 4x² = 3x³ + 7x²

Multiplying Polynomials: Distributive Method

Multiply every term in the first polynomial by every term in the second. That's it.

Multiplying a Monomial by a Polynomial

2x(3x² + 4x - 5) = 2x·3x² + 2x·4x + 2x·(-5) = 6x³ + 8x² - 10x

Multiplying Two Binomials: FOIL Method

FOIL works for (a + b)(c + d):

• First: a × c
• Outer: a × d
• Inner: b × c
• Last: b × d

Example: (x + 3)(x + 5)

First: x × x = x²

Outer: x × 5 = 5x

Inner: 3 × x = 3x

Last: 3 × 5 = 15

Answer: x² + 8x + 15

Multiplying Larger Polynomials

For (x + 2)(x² + 3x - 4), distribute each term:

x(x² + 3x - 4) + 2(x² + 3x - 4)

= x³ + 3x² - 4x + 2x² + 6x - 8

= x³ + 5x² + 2x - 8

Special Products You Should Memorize

These patterns show up constantly:

Commit these to memory. They'll save you time on tests.

Dividing Polynomials

Polynomial Long Division

Divide x² + 5x + 6 by (x + 2):

Step 1: Divide the first term: x² ÷ x = x

Step 2: Multiply: x(x + 2) = x² + 2x

Step 3: Subtract: (x² + 5x) - (x² + 2x) = 3x

Step 4: Bring down the next term: 3x + 6

Step 5: Repeat: 3x ÷ x = 3

Step 6: Multiply: 3(x + 2) = 3x + 6

Step 7: Subtract: (3x + 6) - (3x + 6) = 0

Answer: x + 3

Synthetic Division: Faster Method

Works only when dividing by a linear factor (x - c). Use the opposite sign of the constant.

Divide x² + 5x + 6 by (x - 2):

1. Write coefficients: 1, 5, 6
2. Bring down the 1
3. Multiply by the divisor (2): 1 × 2 = 2. Add to next coefficient: 5 + 2 = 7
4. Multiply 7 by 2: 7 × 2 = 14. Add to next coefficient: 6 + 14 = 20

The remainder is 20. The quotient is x + 7.

When the remainder is 0, the divisor is a factor of the polynomial. That's useful for factoring problems.

Factoring Polynomials

Factoring is breaking down a polynomial into simpler parts that multiply back together. This is essential for solving equations.

Factoring Out the GCF

Find the greatest common factor of all terms:

6x³ + 9x² - 3x

GCF is 3x: 3x(2x² + 3x - 1)

Factoring Trinomials: Reverse FOIL

For x² + 7x + 12:

1. Find two numbers that multiply to 12 (the constant) and add to 7 (the coefficient)
2. 3 and 4 work: 3 × 4 = 12, 3 + 4 = 7
3. Write as (x + 3)(x + 4)

For 2x² + 7x + 3, try different factor combinations of the leading coefficient. This takes practice.

Factoring Difference of Squares

x² - 16 = (x + 4)(x - 4)

4a² - 9b² = (2a + 3b)(2a - 3b)

Factoring by Grouping

For polynomials with four terms, group and factor:

x³ + 3x² + 2x + 6

Group: (x³ + 3x²) + (2x + 6)

Factor each: x²(x + 3) + 2(x + 3)

Factor out (x + 3): (x + 3)(x² + 2)

Polynomial Operations Test: Try These Problems

Test yourself. No calculators.

1. Simplify: (4x² + 3x - 7) + (2x² - 5x + 4)

2. Multiply: (x - 3)(x² + 2x + 5)

3. Factor: 2x² + 5x - 3

4. Divide using synthetic division: (x³ - 6x² + 10x - 3) ÷ (x - 3)

5. Factor completely: 3x³ - 12x

Answers

1. 6x² - 2x - 3

2. x³ - x² - x - 15

3. (2x - 1)(x + 3)

4. x² - 3x + 1 (remainder 0)

5. 3x(x² - 4) = 3x(x + 2)(x - 2)

Common Mistakes to Avoid

• Forgetting to distribute the negative sign when subtracting polynomials
• Not combining all like terms after multiplication
• Skipping signs when factoring trinomials—check every combination
• Using synthetic division when the divisor isn't in the form (x - c)
• Assuming a polynomial can be factored when it can't—sometimes it's already prime

Quick Reference: Operations Summary

Polynomial operations form the backbone of algebra. Once you master addition, multiplication, division, and factoring, you'll handle equations, graphing, and calculus problems without getting stuck on the mechanics.