In order to expand polynomials, we must do the following:
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.
FOIL can be expressed algebraically as:
Using this expression as reference, we can find each of the terms we just described:
\(\textcolor{red}{\text{First}} = (a)(c)\)
\(\textcolor{green}{\text{Outer}} = (a)(d)\)
\(\textcolor{blue}{\text{Inner}} = (b)(c)\)
\(\textcolor{purple}{\text{Last}} = (b)(d)\)
The FOILed expression expands to \(\boldsymbol{ac + ad + bc + bd}\).
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:
\(\textcolor{red}{\text{First}} = (2x)(x) = 2x^2\)
\(\textcolor{green}{\text{Outer}} = (2x)(3) = 6x\)
\(\textcolor{blue}{\text{Inner}} = (1)(x) = x\)
\(\textcolor{purple}{\text{Last}} = (1)(3) = 3\)
We can then simplify the expression by collecting like terms:
\(2x^2 + 6x + x + 3\)
\(2x^2 + 7x + 3\)
Therefore, we can determine using FOIL that \((2x + 1)(x + 3)\) expanded and simplified is \(\boldsymbol{2x^2 + 7x + 3}\).
\((k - 6)^2\)
First, as the expression is multiplying itself, we can rewrite it as such:
Next, we can identify what each of the terms is based on the expression:
\(\textcolor{red}{\text{First}} = (k)(k) = k^2\)
\(\textcolor{green}{\text{Outer}} = (k)(-6) = -6k\)
\(\textcolor{blue}{\text{Inner}} = (-6)(k) = -6k\)
\(\textcolor{purple}{\text{Last}} = (-6)(-6) = 36\)
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 \(\boldsymbol{k^2 - 12k + 36}\).
\((2p - 7q)(2p - 5q)\)
First, we can identify what each of the terms is based on the expression:
\(\textcolor{red}{\text{First}} =(2p)(2p) = 4p^2\)
\(\textcolor{green}{\text{Outer}} = (2p)(-5q) = -10pq\)
\(\textcolor{blue}{\text{Inner}} =(-7q)(2p) = -14pq\)
\(\textcolor{purple}{\text{Last}} = (-7q)(-5q) = 35q^2\)
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 \(\boldsymbol{4p^2 - 24pq + 35q^2}\).
\(-(x - 4)(x - 1) + 5(3x - 1)(2x + 1)\)
First, we can identify what each of the terms is based on the expression, first on the left side:
\(\textcolor{red}{\text{First}}=(x)(x) = x^2\)
\(\textcolor{green}{\text{Outer}}=(x)(-1) = -x\)
\(\textcolor{blue}{\text{Inner}}=(-4)(x) = -4x\)
\(\textcolor{purple}{\text{Last}} =(-4)(-1) = 4\)
Then, we can simplify the expression on the right side:
\(\textcolor{red}{\text{First}} =(3x)(2x) = 6x^2\)
\(\textcolor{green}{\text{Outer}} =(3x)(1) = 3x\)
\(\textcolor{blue}{\text{Inner}} =(-1)(2x) = -2x\)
\(\textcolor{purple}{\text{Last}} =(-1)(1) = -1\)
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 \(\boldsymbol{29x^2 + 10x - 9}\).
\((m - 2)^2 - (3m + 2)^2\)
First, as the expression is multiplying itself, we can rewrite it as such:
Next, we can identify what each of the terms is based on the expression, first on the left side:
\(\textcolor{red}{\text{First}} =(m)(m) = m^2\)
\(\textcolor{green}{\text{Outer}} =(m)(-2) = -2m\)
\(\textcolor{blue}{\text{Inner}} =(-2)(m) = -2m\)
\(\textcolor{purple}{\text{Last}} =(-2)(-2) = 4\)
Then, we can simplify the expression on the right side:
\(\textcolor{red}{\text{First}} =(3m)(3m) = 9m^2\)
\(\textcolor{green}{\text{Outer}} =(3m)(2) = 6m\)
\(\textcolor{blue}{\text{Inner}} =(2)(3m) = 6m\)
\(\textcolor{purple}{\text{Last}} =(2)(2) = 4\)
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 \(\boldsymbol{-8m^2 - 16m}\).
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:
\(= (3x + 4)(x + 2)\)
\(= 3x² + 6x + 4x + 8\)
\(= 3x² + 10x + 8\)
Therefore, the expression \((3x + 4)(x + 2)\) expanded and simplified is \(\boldsymbol{3x² + 10x + 8}\).
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:
\(= 8x^4 + 24x^3 + 2x^3 + 12x^2 + 6x^2 + 10x^2 + 3x + 30x + 15\)
\(= 8x^4 + 26x^3 + 28x^2 + 33x + 15\)
Therefore, the expression \((2x^2 + 6x + 3)(4x^2 + x+ 5)\) expanded and simplified is \(\boldsymbol{8x^4 + 26x^3 + 28x^2 + 33x + 15}\).
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 can determine the area for the white region:
\(A_{\text{White}} = (2x)(x + 1)\)
\(A_{\text{White}} = 2x^2 + 2x\)
Next, we need to determine the area for the entire polynomial:
\(A_{\text{Polynomial}} = (x - 3)(x + 6)\)
\(A_{\text{Polynomial}} = x^2 + 6x - 3x - 18\)
\(A_{\text{Polynomial}} = x^2 + 3x - 18\)
Now that we have the areas of both regions, we can determine the area of the shaded region by calculating the difference between the entire polynomial and the white region:
\(A_{\text{Shaded}} = A_\text{Polynomial} - A_\text{White}\)
\(A_{\text{Shaded}} = x^2 + 3x - 18 - (2x^2 + 2x)\)
\(A_{\text{Shaded}} = x^2 + 3x - 18 - 2x^2 - 2x\)
\(A_{\text{Shaded}} = -x^2 + x - 18\)
Therefore, we can determine that the area of the shaded region is \(\boldsymbol{-x^2 + x - 18}\)