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9.3: Special Products of Polynomials

Difficulty Level: At Grade Created by: CK-12
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Learning Objectives

At the end of this lesson, students will be able to:

  • Find the square of a binomial.
  • Find the product of binomials using sum and difference formula.
  • Solve problems using special products of polynomials.


Terms introduced in this lesson:

second-degree trinomial
square of a binomial, binomial square
sum and difference of terms
difference of squares

Teaching Strategies and Tips

In this lesson, students learn about special products of binomials.

  • Have students learn to recognize the basic patterns.
  • In classroom examples, use colors to denote the numbers playing the role of \begin{align*}a\end{align*}a and \begin{align*}b\end{align*}b in the formulas.

In the special formulas, point out that \begin{align*}b\end{align*}b is considered positive; the sign does not go with the term.


Square the binomial.

\begin{align*}(x - 3)^2\end{align*}(x3)2


The minus sign tells us to use \begin{align*}(a - b)^2 = a^2 - 2ab + b^2\end{align*}(ab)2=a22ab+b2. Setting \begin{align*}a = x\end{align*}a=x and \begin{align*}b = 3\end{align*}b=3 (and not \begin{align*}-3\end{align*}3),

\begin{align*}x^2 - 2(x) (3) + (3)^2 = x^2 - 6x + 9\end{align*}x22(x)(3)+(3)2=x26x+9

Error Troubleshooting

General Tip: Students commit a very common error when they write, for example, \begin{align*}(x + y)^2 = x^2 + y^2\end{align*}(x+y)2=x2+y2; that is, they distribute the exponent over addition instead of multiplication.

  • The power rule for products does not apply to sums or differences within the parentheses. In general, \begin{align*}(x^n + y^m)^p \neq x^{np} + y^{mp}\end{align*}(xn+ym)pxnp+ymp.
  • Students can learn to avoid this mistake by recalling the exponent definition. Therefore, a polynomial raised to an exponent means that the polynomial is multiplied by itself as many times as the exponent indicates. For example:

\begin{align*}(x + y)^2 = (x + y)(x + y)\end{align*}(x+y)2=(x+y)(x+y)

  • See Review Questions 1-4, especially Problem 4.

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