Adding, Subtracting and Multiplying Polynomials
To add or subtract polynomials, combine like terms; to multiply them, multiply every term of one polynomial by every term of the other and then combine like terms. In brief: (2x2 + 3x − 1) + (x2 − 5x + 4) = 3x2 − 2x + 3, and (x + 2)(x2 − x + 3) = x3 + x2 + x + 6.
These three operations are the core of polynomial algebra. The results are always polynomials again: the set of polynomials is closed under addition, subtraction and multiplication. This guide explains each operation with a vertical layout, a table (grid) method for products and a rule for the degree of the result.
How do you add polynomials?
Remove the brackets, then group and add the like terms, which are terms with the same power of the variable. Only the coefficients are added; the exponents stay.
Example 1
(2x2 + 3x − 1) + (x2 − 5x + 4). Group: (2x2 + x2) + (3x − 5x) + (−1 + 4) = 3x2 − 2x + 3.
A vertical layout helps with longer polynomials. Align by powers and fill the gaps with zeros:
x3 + 0x2 + 2x − 5
+ 4x2 − x + 7
= x3 + 4x2 + x + 2
How do you subtract polynomials?
Subtracting means adding the opposite: change the sign of every term in the second polynomial, then add.
Example 2
(5x2 − 2x + 6) − (3x2 + x − 4) = 5x2 − 2x + 6 − 3x2 − x + 4 = 2x2 − 3x + 10. Notice that −(−4) became +4, and the sign of every term in the second bracket changed.
How do you multiply polynomials?
- Monomial times polynomial: distribute: 3x(2x2 − 4x + 5) = 6x3 − 12x2 + 15x.
- Binomial times binomial: multiply each term of the first by each term of the second (the FOIL pattern): (x + 3)(x − 5) = x2 − 5x + 3x − 15 = x2 − 2x − 15.
- General case: multiply every term of one by every term of the other, then collect like terms.
Example 3: binomial times trinomial
(x + 2)(x2 − x + 3) = x · x2 + x · (−x) + x · 3 + 2 · x2 + 2 · (−x) + 2 · 3 = x3 − x2 + 3x + 2x2 − 2x + 6 = x3 + x2 + x + 6. Check with x = 1: (3)(3) = 9 and 1 + 1 + 1 + 6 = 9 ✓.
Example 3b: with a coefficient
(2x − 3)(4x2 + 6x + 9) = 8x3 + 12x2 + 18x − 12x2 − 18x − 27 = 8x3 − 27. This is a difference of cubes.
What is the degree of a sum and of a product?
For nonzero polynomials, the degree of a product is the sum of the degrees: deg(fg) = deg f + deg g. The degree of a sum is at most the larger of the two degrees, and it can be lower when leading terms cancel: (x3 + x) − (x3 − 2) = x + 2 has degree 1.
These rules give a quick check. In Example 3, deg = 1 + 2 = 3 ✓. The leading coefficient of a product is the product of the leading coefficients.
Multiplication grid for (x + 2)(x² − x + 3)
The grid method writes all partial products in a table, which prevents missing terms.
| × | x2 | −x | +3 |
|---|---|---|---|
| x | x3 | −x2 | 3x |
| +2 | 2x2 | −2x | 6 |
Adding the entries by powers gives x3 + (−x2 + 2x2) + (3x − 2x) + 6 = x3 + x2 + x + 6.
Where do these rules come from?
Rules for multiplying expressions with several terms were known in geometric form to Greek mathematicians, who treated products as areas of rectangles, as in Euclid’s Elements. The grid method resembles this area picture. Modern symbolic algebra, developed from the sixteenth century on, expressed them as laws of arithmetic: the distributive, commutative and associative laws, which apply to polynomials as to numbers.
Where are they used today?
Expanding products gives formulas for area and volume: a rectangle with sides x + 3 and x + 5 has area x2 + 8x + 15. Computer algebra systems and spreadsheets do polynomial arithmetic automatically, and algorithms for fast multiplication of very large numbers use the idea that numbers are polynomials evaluated at a base such as 10. Combining like terms is also a first step in simplifying any algebraic expression.
Which mistakes are common?
Not changing all signs in subtraction
−(3x2 + x − 4) = −3x2 − x + 4. Every term changes sign.
Adding unlike terms
2x2 + 3x cannot be combined. Only terms with the same power of x are added.
Multiplying exponents instead of adding them
x2 · x3 = x5, not x6.
Forgetting a product
A binomial times a trinomial has 2 · 3 = 6 products before combining. If you have fewer, you missed some. Test the result with a number such as x = 1.