Adding, Subtracting and Multiplying Polynomials

In this lesson you will: Add and subtract polynomials by combining like terms; Distribute a subtraction across every term of a polynomial; Multiply a monomial or a binomial by a polynomial; Expand a squared binomial without losing the middle term.

Only like terms combine

Like terms have the same variable raised to the same power. 3x^2 and 5x^2 are like terms; 3x^2 and 5x are not, and no amount of wishing makes them one term.

Adding polynomials is therefore easy: gather the x^2 terms, gather the x terms, gather the constants. Subtracting is the same job with one extra step, and that step is where the marks are lost.

Every term times every term

To multiply two polynomials, each term of the first multiplies each term of the second. For (x+3)(x+2) that is four small products, and then you collect what matches.

A box keeps track of them. Two terms times two terms fills a two-by-two grid, so a missing product is visible as an empty cell rather than invisible as a forgotten thought.

Squaring a binomial

(a+b)^2 means (a+b)(a+b), so the box has four cells: a^2, ab, ab and b^2. Two of them are the same, which is where the 2ab comes from.

Draw a square of side a+b and cut it at the point where a ends. You get one a-by-a square, one b-by-b square, and two identical rectangles between them. The two rectangles are the part that gets forgotten.

Common mistake — Squaring the terms instead of the binomial

A student writes (x+5)^2=x^2+25.

The pull is strong, because squaring does distribute over multiplication: (3x)^2 really is 9x^2. The eye generalizes that to addition without being asked.

It fails because (x+5)^2 is (x+5)(x+5), which produces four products, not two. The missing 10x is the pair of rectangles in the square diagram, and it is not a small correction — at x=2 the true value is 49 and the wrong version gives 29.

The same slip appears with subtraction: (x-3)^2 is x^2-6x+9, never x^2-9.

How to catch it: Test with two easy numbers. Take a=3 and b=4: (3+4)^2=49, while 3^2+4^2=25. Any time you are tempted to square term by term, run those numbers — 49 != 25 settles it permanently.