Polynomial Operations
Adding, subtracting, and multiplying polynomials
Polynomials are algebraic expressions made from numbers, variables, and whole-number exponents. Once you learn how to combine like terms and use the distributive property, adding, subtracting, and multiplying polynomials becomes a clear step-by-step process.
What Is a Polynomial?
A polynomial is an expression such as 4x2 + 3x − 7. Its terms are separated by plus or minus signs. The number multiplying a variable is called a coefficient, and the highest exponent is the degree.
Key Rule
Like terms have the same variables raised to the same powers.
Like Terms and Unlike Terms
Terms such as 3x2 and −5x2 are like terms because their variable parts match. However, 3x2 and 3x are unlike terms because their exponents are different.
| Terms | Like or unlike? | Reason |
|---|---|---|
| 4x and −9x | Like | Both have x1. |
| 2x2 and 2x | Unlike | The exponents are different. |
| 7 and −3 | Like | Both are constants. |
On a phone, swipe left or right inside the table if needed.
Adding Polynomials
To add polynomials, remove the parentheses and combine like terms. Keep each term’s sign attached to it.
Worked Example 1
Add (3x2 + 4x − 5) and (2x2 − x + 9).
3x2 + 4x − 5 + 2x2 − x + 9
= 5x2 + 3x + 4
Subtracting Polynomials
When subtracting a polynomial, distribute the negative sign to every term inside the second set of parentheses. This is the step that prevents most sign errors.
Worked Example 2
Subtract (x2 − 3x + 2) from (5x2 + x − 7).
5x2 + x − 7 − x2 + 3x − 2
= 4x2 + 4x − 9
Memory Tip for Subtraction
Change subtraction into addition of the opposite. For example, subtracting x2 − 3x + 2 changes it to −x2 + 3x − 2.
Multiplying Monomials
Before multiplying full polynomials, remember the monomial rule: multiply the coefficients and add the exponents of matching variables.
Multiply 3 × 4 and add 2 + 3.
Multiplying Binomials
Use the distributive property to multiply every term in the first polynomial by every term in the second. The FOIL memory aid works for two binomials: First, Outside, Inside, Last.
Worked Example 3
Multiply (x + 3)(x + 5).
x × x + x × 5 + 3 × x + 3 × 5
= x2 + 5x + 3x + 15 = x2 + 8x + 15
Multiplying a Polynomial by a Binomial
Worked Example 4
Multiply 2x(x2 − 4x + 3):
2x × x2 − 2x × 4x + 2x × 3
= 2x3 − 8x2 + 6x
Special Product Patterns
Some products can be recognized quickly and are useful for factoring and simplifying expressions.
| Pattern | Identity |
|---|---|
| Square of a sum | (a + b)2 = a2 + 2ab + b2 |
| Square of a difference | (a − b)2 = a2 − 2ab + b2 |
| Difference of squares | (a + b)(a − b) = a2 − b2 |
On a phone, swipe horizontally inside the table if needed.
Common Mistakes to Avoid
Combining unlike terms: x and x2 cannot be combined.
Missing a negative sign: When subtracting parentheses, change every sign inside.
Skipping a product: When multiplying polynomials, every term must multiply every term.
Forgetting exponent rules: When multiplying matching variables, add their exponents.
Practice Problems
1. Simplify: (4x2 + 3x − 2) + (x2 − 5x + 6).
2. Simplify: (6x2 − x + 4) − (2x2 + 3x − 1).
3. Multiply: (x + 4)(x + 2).
4. Multiply: 3x(2x2 − x + 5).
Tap to reveal the answers
1. 5x2 − 2x + 4. 2. 4x2 − 4x + 5. 3. x2 + 6x + 8. 4. 6x3 − 3x2 + 15x.
Final Takeaway
For addition and subtraction, line up like terms and watch the signs. For multiplication, use the distributive property so every term is included. Once you combine like terms and simplify, your polynomial answer is complete.
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