Polynomial Operations

Add and subtract by collecting like terms. Multiply by making sure every pair meets exactly once.

Draw a rectangle x + 3 wide and x + 2 tall. What is its area?

Its area is (x + 3)(x + 2) — that is just what area means. But you can also cut it up.

a rectangle x + 3 wide and x + 2 tall x² 3x 2x 6 x 3 x 2 four patches, four products

The two cuts make four patches, and each patch is easy: x · x, 3 · x, 2 · x and 3 · 2.

Add them: x2 + 3x + 2x + 6 = x2 + 5x + 6.

Nothing can be missed, because no part of a rectangle is left uncovered.

That is all FOIL ever meant — it just named the four patches in an order. The picture keeps working for bigger brackets, where FOIL does not: a 3 × 2 grid has six patches, and you fill in all six.

Multiplying two binomials with an area model a rectangle turns multiplying brackets into adding areas x3 x2 x^23x 2x6 (x + 3)(x + 2) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6 every pair gets multiplied once, which is all that FOIL is really saying

Adding and subtracting

Add by collecting like terms. (3x2 + 5x) + (2x2 - 8x) = 5x2 - 3x.

Subtract by distributing the minus sign to every term in the second polynomial, then collecting.

(4x2 - 3x + 1) - (x2 + 5x - 6) = 4x2 - 3x + 1 - x2 - 5x + 6 = 3x2 - 8x + 7.

The minus reaches everything

Subtracting (x2 + 5x - 6) gives -x2 - 5x + 6. All three signs change, including the one that was already negative.

Multiplying

Every term in the first bracket must multiply every term in the second. For two binomials that is four products — the origin of FOIL.

For bigger brackets, FOIL does not apply, but the principle does. Use a grid and fill in every cell.

Then collect like terms and write the answer in standard form.

Use a box, not a mnemonic

Draw a rectangle with one polynomial along the top and the other down the side. Fill every cell, then add. It never misses a product, whatever the sizes.

Special products worth knowing

(a + b)2 = a2 + 2ab + b2. The middle term is easy to forget.

(a - b)2 = a2 - 2ab + b2.

(a + b)(a - b) = a2 - b2 — the middle terms cancel, leaving a difference of squares.

(x + 3)2 is not x2 + 9

It is (x+3)(x+3) = x2 + 6x + 9. Write the bracket out twice. Squaring a sum is not squaring each piece.

Closure

Add, subtract or multiply two polynomials and you always get a polynomial. That is what closure means.

Division breaks it: x2 + 1x is not a polynomial, because of the 1x.

So polynomials are closed under addition, subtraction and multiplication, but not division — exactly like the integers.

Worked examples

Subtracting

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

  1. Distribute the minus: 5x2 - 2x + 7 - 2x2 - 4x + 3 Every sign in the second bracket flips.
  2. x2 terms: 5x2 - 2x2 = 3x2 Collect like terms.
  3. x terms: -2x - 4x = -6x Both negative now.
  4. Constants: 7 + 3 = 10 -(-3) became +3.

3x2 - 6x + 10

Multiplying binomials

Expand (2x - 5)(3x + 4).

  1. 2x · 3x = 6x2 First times first.
  2. 2x · 4 = 8x and -5 · 3x = -15x The two middle products.
  3. -5 · 4 = -20 Last times last.
  4. 6x2 + 8x - 15x - 20 = 6x2 - 7x - 20 Collect the middle terms.

6x2 - 7x - 20

A special product

Expand (3x - 4)2.

  1. Write it out: (3x - 4)(3x - 4) Never square the terms separately.
  2. 9x2 from the first pair. 3x · 3x.
  3. Middle: -12x - 12x = -24x Twice the product of the two terms.
  4. Last: 16 (-4)(-4) = +16.

9x2 - 24x + 16

Words to know

Polynomial
A sum of terms, each a number times a whole-number power of the variable.
Like terms
Terms with identical variable parts. 4x2 and -9x2 are like; 4x2 and 4x are not.
Binomial
A polynomial with two terms.
Closure
Polynomials added, subtracted or multiplied always give another polynomial.
Standard form
Terms written in order of decreasing degree.

Mistakes to watch for

Distributing a subtraction to only the first term. -(x2 + 5x - 6) is -x2 - 5x + 6: all three signs change, including the one that was already negative. This is the most reliable source of errors in the unit.

Writing (x + 5)2 as x2 + 25. Squaring a sum is not squaring each piece — write the bracket out twice and the middle term 10x appears. Check with x = 1: 36, not 26.

Adding 3x2 and 4x as like terms. They are not: x2 and x are different quantities, the way an area and a length are. Only identical variable parts can be collected.

Missing a product when the brackets get bigger. FOIL only has four letters and stops working past two binomials. Draw the grid and fill every cell — it never misses.

Algebra tiles — physical squares and rectangles for x2, x and 1 — became a standard classroom tool in the 1960s.

But the idea is far older: Euclid's Book II proves algebraic identities entirely with rectangles, because the Greeks had no algebraic notation.

His Proposition 4 is (a+b)2 = a2 + 2ab + b2, stated and proved as a fact about squares on a line.

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