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Study Guides > College Algebra

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.
 (x+5)2= x2+10x+25(x−3)2= x2−6x+9(4x−1)2=4x2−8x+1\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}
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.
(x+a)2=(x+a)(x+a)=x2+2ax+a2{\left(x+a\right)}^{2}=\left(x+a\right)\left(x+a\right)={x}^{2}+2ax+{a}^{2}

How To: Given a binomial, square it using the formula for perfect square trinomials.

  1. Square the first term of the binomial.
  2. Square the last term of the binomial.
  3. For the middle term of the trinomial, double the product of the two terms.
  4. Add and simplify.

Example: Expanding Perfect Squares

Expand (3x−8)2{\left(3x - 8\right)}^{2}.

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.

(3x)2−2(3x)(8)+(−8)2{\left(3x\right)}^{2}-2\left(3x\right)\left(8\right)+{\left(-8\right)}^{2}

9x2−48x+649{x}^{2}-48x+64.

Try It

Expand (4x−1)2{\left(4x - 1\right)}^{2}.

Answer: 16x2−8x+116{x}^{2}-8x+1

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:
(a+2b)(4a−b−c)a(4a−b−c)+2b(4a−b−c)Use the distributive property.4a2−ab−ac+8ab−2b2−2bcMultiply.4a2+(−ab+8ab)−ac−2b2−2bcCombine like terms.4a2+7ab−ac−2bc−2b2Simplify.\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}

Example: Multiplying Polynomials Containing Several Variables

Multiply (x+4)(3x−2y+5)\left(x+4\right)\left(3x - 2y+5\right).

Answer: Follow the same steps that we used to multiply polynomials containing only one variable.

x(3x−2y+5)+4(3x−2y+5)Use the distributive property.3x2−2xy+5x+12x−8y+20Multiply.3x2−2xy+(5x+12x)−8y+20Combine like terms.3x2−2xy+17x−8y+20Simplify.\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}

Try It

(3x−1)(2x+7y−9)\left(3x - 1\right)\left(2x+7y - 9\right).

Answer: 6x2+21xy−29x−7y+96{x}^{2}+21xy - 29x - 7y+9

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