Polynomial operations (add, subtract, multiply)
Polynomial Operations
Polynomial Operations is the opening topic of Polynomial Functions in the Common Core State Standards. It is aligned to Standard A-APR.1, which requires students to add, subtract, and multiply polynomials, understanding they are closed under these operations.
Adding, subtracting, and multiplying polynomials combines like terms and distributes every pair of terms.
Theory
A polynomial is a sum of terms \(a_nx^n+\dots+a_1x+a_0\). To operate on polynomials:
- Add / subtract: combine like terms (same power of \(x\)).
- Subtract by distributing the minus sign first.
- Multiply by distributing every term of one factor across the other.
Multiplying and combining:
How to operate on polynomials
- For adding/subtracting, align like terms.
- Distribute a subtraction sign to every term.
- For products, distribute each term, then combine.
- Write the result in descending order of degree.
Combine like terms.
| \((3x^2+x^2)+(-2x+4x)+(5-1)\) | ||
| \(=\) | \(4x^2+2x+4\) |
Distribute the minus sign, then combine.
| \(2x^2+3x-4-x^2+x-6\) | ||
| \(=\) | \(x^2+4x-10\) |
Multiply each pair (FOIL) and combine.
| \(x^2+2x+3x+6\) | ||
| \(=\) | \(x^2+5x+6\) |
Distribute each term of the first factor.
| \(x^3-3x^2+2x+x^2-3x+2\) | ||
| \(=\) | \(x^3-2x^2-x+2\) |
Common pitfalls
Frequently asked questions
How do you add polynomials?
Combine like terms — terms with the same power of \(x\).
How do you subtract polynomials?
Distribute the minus sign, then combine like terms.
How do you multiply polynomials?
Distribute every term of one factor across the other, then combine.
What is the degree of a product?
The sum of the degrees of the factors.