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
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:
\((k - 6)^2\)
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First, as the expression is multiplying itself, we can rewrite it as such:
\((k - 6)(k - 6)\)
Next, we can identify what each of the terms is based on the expression:
First\( = (k)(k) = k^2\)
Outer\( = (k)(-6) = -6k\)
Inner\( = (-6)(k) = -6k\)
Last\( = (-6)(-6) = 36\)
We can then simplify the expression by collecting like terms:
\(= k^2 - 6k - 6k + 36\)
\(= k^2 - 12k + 36\)
Therefore, we can determine that \((k - 6)^2\) expanded and simplified is \(k^2 - 12k + 36\).
\((2p - 7q)(2p - 5q)\)
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First, we can identify what each of the terms is based on the expression:
First \( =(2p)(2p) = 4p^2\)
Outer \( =(2p)(-5q) = -10pq\)
Inner \( =(-7q)(2p) = -14pq\)
Last \( =(-7q)(-5q) = 35q^2\)
We can then simplify the expression by collecting like terms:
\(= 4p^2 - 10pq - 14pq + 35q^2\)
\(= 4p^2 - 24pq + 35q^2\)
Therefore, we can determine that \((2p - 7q)(2p - 5q)\) expanded and simplified is \(4p^2 - 24pq + 35q^2\).
\(-(x - 4)(x - 1) + 5(3x - 1)(2x + 1)\)
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First, we can identify what each of the terms is based on the expression, first on the left side:
First \( =(x)(x) = x^2\)
Outer \( =(x)(-1) = -x\)
Inner \( =(-4)(x) = -4x\)
Last \( =(-4)(-1) = 4\)
Then, we can simplify the expression on the right side:
First \( =(3x)(2x) = 6x^2\)
Outer \( =(3x)(1) = 3x\)
Inner \( =(-1)(2x) = -2x\)
Last \( =(-1)(1) = -1\)
We can simplify the expression by expanding and collecting like terms:
\(= -(x^2 - x - 4x + 4) + 5(6x^2 + 3x - 2x - 1)\)
\(= -(x^2 - 5x + 4) + 5(6x^2 + x - 1)\)
\(= -x^2 + 5x - 4 + 30x^2 + 5x - 5)\)
\(= 29x^2 + 10x - 9)\)
Therefore, we can determine that \(-(x - 4)(x - 1) + 5(3x - 1)(2x + 1)\) expanded and simplified is \(29x^2 + 10x - 9\).
\((m - 2)^2 - (3m + 2)^2\)
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First, as the expression is multiplying itself, we can rewrite it as such:
\((m - 2)(m - 2) - (3m + 2)(3m + 2)\)
Next, we can identify what each of the terms is based on the expression, first on the left side:
First \( =(m)(m) = m^2\)
Outer \( =(m)(-2) = -2m\)
Inner \( =(-2)(m) = -2m\)
Last \( =(-2)(-2) = 4\)
Then, we can simplify the expression on the right side:
First \( =(3m)(3m) = 9m^2\)
Outer \( =(3m)(2) = 6m\)
Inner \( =(2)(3m) = 6m\)
Last \( =(2)(2) = 4\)
We can simplify the expression by expanding and collecting like terms:
\(= m^2 - 2m - 2m + 4 - (9m^2 + 6m + 6m + 4)\)
\(= m^2 - 4m + 4 - (9m^2 + 12m + 4)\)
\(= m^2 - 4m + 4 - 9m^2 - 12m - 4\)
\(= -8m^2 - 16m\)
Therefore, we can determine that \((m - 2)^2 - (3m + 2)^2\) expanded and simplified is \(-8m^2 - 16m\).
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:
Show Answer
In order to determine the area of the shaded region, we need to subtract the area of the white region from the entire polynomial.
First, we need to determine the areas for the white region and the entire polynomial:
\(\text{White Region} = (2x)(x + 1)\)
\(\text{White Region} = 2x^2 + 2x\)
\(\text{Polynomial} = (x - 3)(x + 6)\)
\(\text{Polynomial} = x^2 + 6x - 3x - 18\)
\(\text{Polynomial} = x^2 + 3x - 18\)
Now that we have the areas of both regions, we can get the area of the shaded region by subtracting the area of the white region from the entire polynomial:
\(\text{Area} = \text{Polynomial} - \text{White Region}\)
\(\text{Area} = x^2 + 3x - 18 - (2x^2 + 2x)\)
\(\text{Area} = x^2 + 3x - 18 - 2x^2 - 2x\)
\(\text{Area} = -x^2 + x - 18\)
Therefore, we can determine that the area of the shaded region is \(-x^2 + x - 18\)