Special Cases of Polynomials
Learning Objectives
- Square a binomial
- Find a difference of squares
- Perform operations on polynomials with several variables
Perfect Square Trinomials
Certain binomial products have special forms. When a binomial is squared, the result is called a perfect square trinomial. We can find the square by multiplying the binomial by itself. However, there is a special form that each of these perfect square trinomials takes, and memorizing the form makes squaring binomials much easier and faster. Let’s look at a few perfect square trinomials to familiarize ourselves with the form.[latex]\begin{array}{ccc}\hfill \text{ }{\left(x+5\right)}^{2}& =& \text{ }{x}^{2}+10x+25\hfill \\ \hfill {\left(x - 3\right)}^{2}& =& \text{ }{x}^{2}-6x+9\hfill \\ \hfill {\left(4x - 1\right)}^{2}& =& 4{x}^{2}-8x+1\hfill \end{array}[/latex]
Notice that the first term of each trinomial is the square of the first term of the binomial and, similarly, the last term of each trinomial is the square of the last term of the binomial. The middle term is double the product of the two terms. Lastly, we see that the first sign of the trinomial is the same as the sign of the binomial.
A General Note: Perfect Square Trinomials
When a binomial is squared, the result is the first term squared added to double the product of both terms and the last term squared.[latex]{\left(x+a\right)}^{2}=\left(x+a\right)\left(x+a\right)={x}^{2}+2ax+{a}^{2}[/latex]
How To: Given a binomial, square it using the formula for perfect square trinomials.
- Square the first term of the binomial.
- Square the last term of the binomial.
- For the middle term of the trinomial, double the product of the two terms.
- Add and simplify.
Example: Expanding Perfect Squares
Expand [latex]{\left(3x - 8\right)}^{2}[/latex].Answer: Begin by squaring the first term and the last term. For the middle term of the trinomial, double the product of the two terms.
[latex]{\left(3x\right)}^{2}-2\left(3x\right)\left(8\right)+{\left(-8\right)}^{2}[/latex]
[latex]9{x}^{2}-48x+64[/latex].
Try It
Expand [latex]{\left(4x - 1\right)}^{2}[/latex].Answer: [latex]16{x}^{2}-8x+1[/latex]
Performing Operations with Polynomials of Several Variables
We have looked at polynomials containing only one variable. However, a polynomial can contain several variables. All of the same rules apply when working with polynomials containing several variables. Consider an example:[latex]\begin{array}{cc}\left(a+2b\right)\left(4a-b-c\right)\hfill & \hfill \\ a\left(4a-b-c\right)+2b\left(4a-b-c\right)\hfill & \text{Use the distributive property}.\hfill \\ 4{a}^{2}-ab-ac+8ab - 2{b}^{2}-2bc\hfill & \text{Multiply}.\hfill \\ 4{a}^{2}+\left(-ab+8ab\right)-ac - 2{b}^{2}-2bc\hfill & \text{Combine like terms}.\hfill \\ 4{a}^{2}+7ab-ac - 2bc - 2{b}^{2}\hfill & \text{Simplify}.\hfill \end{array}[/latex]
Example: Multiplying Polynomials Containing Several Variables
Multiply [latex]\left(x+4\right)\left(3x - 2y+5\right)[/latex].Answer: Follow the same steps that we used to multiply polynomials containing only one variable.
[latex]\begin{array}{cc}x\left(3x - 2y+5\right)+4\left(3x - 2y+5\right) \hfill & \text{Use the distributive property}.\hfill \\ 3{x}^{2}-2xy+5x+12x - 8y+20\hfill & \text{Multiply}.\hfill \\ 3{x}^{2}-2xy+\left(5x+12x\right)-8y+20\hfill & \text{Combine like terms}.\hfill \\ 3{x}^{2}-2xy+17x - 8y+20 \hfill & \text{Simplify}.\hfill \end{array}[/latex]
Try It
[latex]\left(3x - 1\right)\left(2x+7y - 9\right)[/latex].Answer: [latex]6{x}^{2}+21xy - 29x - 7y+9[/latex]
Licenses & Attributions
CC licensed content, Original
- Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.
CC licensed content, Shared previously
- College Algebra. Provided by: OpenStax Authored by: Abramson, Jay et al.. License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/[email protected].
- Question ID 1825. Authored by: David Whittaker. License: CC BY: Attribution. License terms: IMathAS Community License CC- BY + GPL.
- Question 1856. Authored by: Lawrence Morales. License: CC BY: Attribution. License terms: IMathAS Community License CC- BY + GPL.
- Question ID 100774. Authored by: Rick Rieman. License: CC BY: Attribution. License terms: IMathAS Community License CC- BY + GPL.
CC licensed content, Specific attribution
- College Algebra. Provided by: OpenStax Authored by: OpenStax College Algebra. Located at: https://cnx.org/contents/[email protected]:1/Preface. License: CC BY: Attribution.
