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

- Multiply each term in one polynomial by each term in the other polynomial(s)
- 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** 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:

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 \(ac + ad + bc + bd\).

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:

Outer \(= (2x)(3) = 6x\)

Inner \(= (1)(x) = x\)

Last \(= (1)(3) = 3\)

We can then simplify the expression by collecting like terms:

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:

\((k - 6)^2\)

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

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

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

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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:

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:

\(= 3x² + 6x + 4x + 8\)

\(= 3x² + 10x + 8\)

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

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:

\(= 8x^4 + 26x^3 + 28x^2 + 33x + 15\)

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:

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