# Expanding Polynomials

In order to expand polynomials, we must do the following:

1. Multiply each term in one polynomial by each term in the other polynomial(s)
2. Simplify the expression by collecting like terms

There are a couple of different techniques covered in this lesson that can help us expand polynomials.

## FOIL

FOIL is a helpful acronym which stands for First, Outer, Inner, Last.
It represents the order that we multiply terms between 2 binomials.
We first multiply the First terms in each binomial, followed in respective order by the Outer, Inner and Last terms.

It can be expressed algebraically as:

$$(a + b)(c + d)$$

Using this expression as reference, we can find each of the terms we just described:

• First $$= (a)(c)$$
• Outer $$= (a)(d)$$
• Inner $$= (b)(c)$$
• Last $$= (b)(d)$$

The FOILed expression expands to $$ac + ad + bc + bd$$.

Example

Expand and simplify the expression $$(2x + 1)(x + 3)$$ using FOIL.

First, we can determine what each of the terms is based on the expression:

First $$= (2x)(x) = 2x^2$$
Outer $$= (2x)(3) = 6x$$
Inner $$= (1)(x) = x$$
Last $$= (1)(3) = 3$$

We can then simplify the expression by collecting like terms:

$$2x^2 + 7x + 3$$

Therefore, we can determine using FOIL that $$(2x + 1)(x + 3)$$ expanded and simplified is $$2x^2 + 7x + 3$$.

Expand and simplify the following polynomials:

$$(k - 6)^2$$

$$(2p - 7q)(2p - 5q)$$

$$-(x - 4)(x - 1) + 5(3x - 1)(2x + 1)$$

$$(m - 2)^2 - (3m + 2)^2$$

## Picture Method

The Picture Method is a helpful way of multiplying polynomials visually instead of algebraically. This is very similar to finding the Area of a square or rectangle.
Imagine each term of the binomials like the dimensions of a wall. Each multiplied term would represent the different bricks used to build that wall:

Example

Expand and simplify the expression $$(3x + 4)(x + 2)$$ using the Picture Method.

We can visualize the dimensions of the expression as such:

We can use these dimensions as the terms for determining the simplified expression:

$$= (3x + 4)(x + 2)$$
$$= 3x² + 6x + 4x + 8$$
$$= 3x² + 10x + 8$$

Therefore, the expression $$(3x + 4)(x + 2)$$ expanded and simplified is $$3x² + 10x + 8$$.

Example

Expand and simplify the expression $$(2x^2 + 6x + 3)(4x^2 + x+ 5)$$.

We can visualize the dimensions of the expression as such:

We can use these dimensions as the terms for determining the simplified expression:

$$= 8x^4 + 24x^3 + 2x^3 + 12x^2 + 6x^2 + 10x^2 + 3x + 30x + 15$$
$$= 8x^4 + 26x^3 + 28x^2 + 33x + 15$$

Therefore, the expression $$(2x^2 + 6x + 3)(4x^2 + x+ 5)$$ expanded and simplified is $$8x^4 + 26x^3 + 28x^2 + 33x + 15$$.

Write and simplify an expression to represent the shaded area of the following polynomial: