Special Products of Polynomials

Finding the Square of a Binomial

Read the information, which gives examples of squaring binomials and the general formulas that always work for them. Example 1 shows how to use the formula. Note the definitions of sum and difference, which are used when multiplying one addition binomial and one subtraction binomial. Example 2 shows how to use this definition. After you have reviewed the material, complete questions 1–5 and 14–18 in the practice set and check your answers.

When we multiply two linear (degree of 1) binomials, we create a quadratic (degree of 2) polynomial with four terms. The middle terms are like terms so we can combine them and simplify to get a quadratic or 2^nddegree trinomial (polynomial with three terms). In this lesson, we will talk about some special products of binomials.

A special binomial product is the square of a binomial. Consider the following multiplication: \(\begin{align*}(x+4)(x+4)\end{align*}\). We are multiplying the same expression by itself, which means that we are squaring the expression. This means that:

\(\begin{align*}(x+4)(x+4) & = (x+4)^2\\ (x+4)(x+4) & = x^2+4x+4x+16=x^2+8x+16\end{align*}\)

This follows the general pattern of the following rule.

Square of a Binomial: \(\begin{align*}(a+b)^2=a^2+2ab+b^2\end{align*}\), and \(\begin{align*}(a-b)^2=a^2-2ab+b^2\end{align*}\)

Stay aware of the common mistake \(\begin{align*}(a+b)^2=a^2+b^2\end{align*}\). To see why \(\begin{align*}(a+b)^2 \neq a^2+b^2\end{align*}\), try substituting numbers for \(\begin{align*}a\end{align*}\) and \(\begin{align*}b\end{align*}\) into the equation (for example, \(\begin{align*}a=4\end{align*}\) and \(\begin{align*}b=3\end{align*}\)), and you will see that it is not a true statement. The middle term, \(\begin{align*}2ab\end{align*}\), is needed to make the equation work.

Example 1: Simplify by multiplying: \(\begin{align*}(x+10)^2\end{align*}\).

Solution: Use the square of a binomial formula, substituting \(\begin{align*}a=x\end{align*}\) and \(\begin{align*}b=10\end{align*}\)

\(\begin{align*}(a+b)^2&=a^2+2ab+b^2\\ (x+10)^2 & =(x)^2+2(x)(10)+(10)^2=x^2+20x+100\end{align*}\)

Source: cK-12, https://www.ck12.org/book/basic-algebra/section/9.3/
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