9.3: Special Products of Polynomials
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
Finding the Square of a Binomial
A special binomial product is the square of a binomial. Consider the following multiplication:
This follows the general pattern of the following rule.
Square of a Binomial:
Stay aware of the common mistake
Example 1: Simplify by multiplying:
Solution: Use the square of a binomial formula, substituting
Finding the Product of Binomials Using Sum and Difference Patterns
Another special binomial product is the product of a sum and a difference of terms. For example, let’s multiply the following binomials.
Notice that the middle terms are opposites of each other, so they cancel out when we collect like terms. This always happens when we multiply a sum and difference of the same terms.
When multiplying a sum and difference of the same two terms, the middle terms cancel out. We get the square of the first term minus the square of the second term. You should remember this formula.
Sum and Difference Formula:
Example 2: Multiply the following binomias and simplify.
Solution: Use the above formula, substituting
Solving RealWorld Problems Using Special Products of Polynomials
Let’s now see how special products of polynomials apply to geometry problems and to mental arithmetic. Look at the following example.
Example: Find the area of the square.
Solution:
Notice that this gives a visual explanation of the square of binomials product.
The next example shows how to use the special products in doing fast mental calculations.
Example 3: Find the products of the following numbers without using a calculator.
(a)
(b)
Solution: The key to these mental “tricks” is to rewrite each number as a sum or difference of numbers you know how to square easily.
(a) Rewrite
Then
(b)
Practice Set
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both.
CK12 Basic Algebra: Special Products of Binomials (10:36)
Use the special product for squaring binomials to multiply these expressions.

(x+9)2 
(x−1)2 
(2y+6)2 
(3x−7)2 
(7c+8)2 
(9a2+6)2 
(b2−1)2 
(m3+4)2 
(14t+2)2 
(6k−3)2 
(a3−7)2 
(4x2+y2)2 
(8x−3)2
Use the special product of a sum and difference to multiply these expressions.

(2x−1)(2x+1) 
(2x−3)(2x+3) 
(4+6x)(4−6x) 
(6+2r)(6−2r) 
(−2t+7)(2t+7) 
\begin{align*}(8z8)(8z+8)\end{align*}
(8z−8)(8z+8) 
\begin{align*}(3x^2+2)(3x^22)\end{align*}
(3x2+2)(3x2−2) 
\begin{align*}(x12)(x+12)\end{align*}
(x−12)(x+12) 
\begin{align*}(5a2b)(5a+2b)\end{align*}
(5a−2b)(5a+2b) 
\begin{align*}(ab1)(ab+1)\end{align*}
(ab−1)(ab+1)
Find the area of the orange square in the following figure. It is the lower right shaded box.
Multiply the following numbers using the special products.

\begin{align*}45\times 55\end{align*}
45×55 
\begin{align*}97 \times 83\end{align*}
97×83  \begin{align*}19^2\end{align*}
 \begin{align*}56^2\end{align*}
 \begin{align*}876 \times 824\end{align*}
 \begin{align*}1002 \times 998\end{align*}
 \begin{align*}36 \times 44\end{align*}
Mixed Review
 Simplify \begin{align*}5x(3x+5)+11(7x)\end{align*}.
 Cal High School has grades nine through twelve. Of the school's student population, \begin{align*}\frac{1}{4}\end{align*} are freshmen, \begin{align*}\frac{2}{5}\end{align*} are sophomores, \begin{align*}\frac{1}{6}\end{align*} are juniors, and 130 are seniors. To the nearest whole person, how many students are in the sophomore class?
 Kerrie is working at a toy store and must organize 12 bears on a shelf. In how many ways can this be done?
 Find the slope between \begin{align*}\left ( \frac{3}{4},1 \right )\end{align*} and \begin{align*}\left ( \frac{3}{4}, 16 \right )\end{align*}.
 If \begin{align*}1 \ lb=454 \ grams\end{align*}, how many kilograms does a 260pound person weigh?
 Solve for \begin{align*}v\end{align*}: \begin{align*}16v=3\end{align*}.
 Is \begin{align*}y=x^4+3x^2+2\end{align*} a function? Use the definition of a function to explain.