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# 9.3: Special Products of Polynomials

Difficulty Level: At Grade Created by: CK-12

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 2nd\begin{align*}2^{nd}\end{align*} degree trinomial (polynomial with three terms). In this lesson, we will talk about some special products of binomials.

## Finding the Square of a Binomial

A special binomial product is the square of a binomial. Consider the following multiplication: (x+4)(x+4)\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:

(x+4)(x+4)(x+4)(x+4)=(x+4)2=x2+4x+4x+16=x2+8x+16\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: (a+b)2=a2+2ab+b2\begin{align*}(a+b)^2=a^2+2ab+b^2\end{align*}, and (ab)2=a22ab+b2\begin{align*}(a-b)^2=a^2-2ab+b^2\end{align*}

Stay aware of the common mistake (a+b)2=a2+b2\begin{align*}(a+b)^2=a^2+b^2\end{align*}. To see why (a+b)2a2+b2\begin{align*}(a+b)^2 \neq a^2+b^2\end{align*}, try substituting numbers for a\begin{align*}a\end{align*} and b\begin{align*}b\end{align*} into the equation (for example, a=4\begin{align*}a=4\end{align*} and b=3\begin{align*}b=3\end{align*}), and you will see that it is not a true statement. The middle term, 2ab\begin{align*}2ab\end{align*}, is needed to make the equation work.

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

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

(a+b)2(x+10)2=a2+2ab+b2=(x)2+2(x)(10)+(10)2=x2+20x+100\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*}

## 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.

(x+4)(x4)=x24x+4x16=x216\begin{align*}(x+4)(x-4) & = x^2-4x+4x-16\\ & = x^2-16\end{align*}

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.

(a+b)(ab)=a2ab+abb2=a2b2\begin{align*}(a+b)(a-b)&=a^2-ab+ab-b^2\\ & =a^2-b^2\end{align*}

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: (a+b)(ab)=a2b2\begin{align*}(a+b)(a-b) = a^2-b^2\end{align*}

Example 2: Multiply the following binomias and simplify.

(5x+9)(5x9)\begin{align*}(5x+9)(5x-9)\end{align*}

Solution: Use the above formula, substituting a=5x\begin{align*}a=5x\end{align*} and b=9\begin{align*}b=9\end{align*}. Multiply.

(5x+9)(5x9)=(5x)2(9)2=25x281\begin{align*}(5x+9)(5x-9)=(5x)^2-(9)^2=25x^2-81\end{align*}

## Solving Real-World 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: The area of the square=side×side\begin{align*}The \ area \ of \ the \ square = side \times side\end{align*}

Area=(a+b)(a+b)=a2+2ab+b2\begin{align*}\text{Area} & = (a+b)(a+b)\\ & = a^2+2ab+b^2\end{align*}

Notice that this gives a visual explanation of the square of binomials product.

Area of big square:(a+b)2=Area of blue square=a2+2 (area of yellow)=2ab+area of red square=b2\begin{align*}Area \ of \ big \ square: (a+b)^2 = Area \ of \ blue \ square = a^2+2 \ (area \ of \ yellow) = 2ab + area \ of \ red \ square = b^2\end{align*}

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) 43×57\begin{align*}43 \times 57\end{align*}

(b) 452\begin{align*}45^2\end{align*}

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 43=(507)\begin{align*}43=(50-7)\end{align*} and 57=(50+7)\begin{align*}57=(50+7)\end{align*}.

Then 43×57=(507)(50+7)=(50)2(7)2=250049=2,451\begin{align*}43 \times 57 = (50-7)(50+7) = (50)^2-(7)^2=2500-49=2,451\end{align*}.

(b) 452=(40+5)2=(40)2+2(40)(5)+(5)2=1600+400+25=2,025\begin{align*}45^2 = (40+5)^2 = (40)^2+2(40)(5) +(5)^2 = 1600+400+25=2,025\end{align*}

## 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.

Use the special product for squaring binomials to multiply these expressions.

1. (x+9)2\begin{align*}(x+9)^2\end{align*}
2. (x1)2\begin{align*}(x-1)^2\end{align*}
3. (2y+6)2\begin{align*}(2y+6)^2\end{align*}
4. (3x7)2\begin{align*}(3x-7)^2\end{align*}
5. (7c+8)2\begin{align*}(7c+8)^2\end{align*}
6. (9a2+6)2\begin{align*}(9a^2+6)^2\end{align*}
7. (b21)2\begin{align*}(b^2-1)^2\end{align*}
8. (m3+4)2\begin{align*}(m^3+4)^2\end{align*}
9. \begin{align*}\left ( \frac{1}{4} t+2 \right )^2\end{align*}
10. \begin{align*}(6k-3)^2\end{align*}
11. \begin{align*}(a^3-7)^2\end{align*}
12. \begin{align*}(4x^2+y^2)^2\end{align*}
13. \begin{align*}(8x-3)^2\end{align*}

Use the special product of a sum and difference to multiply these expressions.

1. \begin{align*}(2x-1)(2x+1)\end{align*}
2. \begin{align*}(2x-3)(2x+3)\end{align*}
3. \begin{align*}(4+6x)(4-6x)\end{align*}
4. \begin{align*}(6+2r)(6-2r)\end{align*}
5. \begin{align*}(-2t+7)(2t+7)\end{align*}
6. \begin{align*}(8z-8)(8z+8)\end{align*}
7. \begin{align*}(3x^2+2)(3x^2-2)\end{align*}
8. \begin{align*}(x-12)(x+12)\end{align*}
9. \begin{align*}(5a-2b)(5a+2b)\end{align*}
10. \begin{align*}(ab-1)(ab+1)\end{align*}

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.

1. \begin{align*}45\times 55\end{align*}
2. \begin{align*}97 \times 83\end{align*}
3. \begin{align*}19^2\end{align*}
4. \begin{align*}56^2\end{align*}
5. \begin{align*}876 \times 824\end{align*}
6. \begin{align*}1002 \times 998\end{align*}
7. \begin{align*}36 \times 44\end{align*}

Mixed Review

1. Simplify \begin{align*}5x(3x+5)+11(-7-x)\end{align*}.
2. 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?
3. Kerrie is working at a toy store and must organize 12 bears on a shelf. In how many ways can this be done?
4. 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*}.
5. If \begin{align*}1 \ lb=454 \ grams\end{align*}, how many kilograms does a 260-pound person weigh?
6. Solve for \begin{align*}v\end{align*}: \begin{align*}|16-v|=3\end{align*}.
7. Is \begin{align*}y=x^4+3x^2+2\end{align*} a function? Use the definition of a function to explain.

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