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# 9.3: Special Products of Polynomials

Difficulty Level: At Grade Created by: CK-12

## 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.

## Vocabulary

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 a\begin{align*}a\end{align*} and b\begin{align*}b\end{align*} in the formulas.

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

Example:

Square the binomial.

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

Solution:

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

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

## Error Troubleshooting

General Tip: Students commit a very common error when they write, for example, (x+y)2=x2+y2\begin{align*}(x + y)^2 = x^2 + y^2\end{align*}; 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, (xn+ym)pxnp+ymp\begin{align*}(x^n + y^m)^p \neq x^{np} + y^{mp}\end{align*}.
• 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:

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

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

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