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Special Products of Polynomials

Work with multiple variables with special products of polynomials. Learn how to solve binomial squares, and the difference of squares with binomials.

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Special Products of Polynomials

Suppose that the entrance to a dog house is a square, with a height of \begin{align*}x\end{align*} inches and a width of \begin{align*}x\end{align*} inches. Because your puppy has grown, you need to increase both the height and the width by 4 inches. What would be the area of the resulting entrance to the dog house? If the height had been increased by 4 inches while the width had been decreased by 4 inches, what would the area have been then? 

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 \begin{align*}2^{nd}\end{align*} degree trinomial (polynomial with three terms). There are some special products of binomials that make finding simplifying expressions and equations easier.

The Square of a Binomial

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.

Let's simplify the following:

\begin{align*}(x+10)^2\end{align*}

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*}

The Product of Binomials with Patterns

Another special binomial product is the product of a sum and a difference of terms. For example, let’s multiply the following binomials.

\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.

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

Let's multiply the following binomials and simplify:

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

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

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

Real-World Problems with Special Products of Polynomials

Let’s now see how special products of polynomials apply to geometry problems and to mental arithmetic. 

Let's find the area of the whole square in the figure above:

\begin{align*}The \ area \ of \ the \ square = side \times side\end{align*}.\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.

\begin{align*}\text{Big square area} &= (a+b)^2 \\ (a+b)^2 & = a^2 (\text{Blue area}) + 2ab (\text{Yellow area}) + b^2 (\text{Red area})\\ (a+b)^2 & =a^2+2ab+b^2\end{align*}

The next example shows how to use the special products in doing fast mental calculations.

Now, let's find the products of the following numbers without using a calculator.

  1. \begin{align*}43 \times 57\end{align*}

The key to these mental “tricks” is to rewrite each number as a sum or difference of numbers you know how to square easily.

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

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

  1. \begin{align*}45^2\end{align*}

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

Examples

Example 1

Earlier, you were told that the entrance to a dog house is square with a height and width of \begin{align*}x\end{align*} inches. Because your puppy has grown, you need to increase both the height and width by 4 inches. What would be the area of the resulting entrance to the dog house? If the height had been increased by 4 inches while the width had been decreased by 4 inches, what would the area have been then?

The area of the entrance is the length multiplied by the height. If both are increased by 4, then the area of the entrance becomes:

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

We can use the Square of a Binomial formula, \begin{align*}(a+b)^2=a^2+2ab+b^2\end{align*}, to simplify

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

The area of the doghouse if both the height and width are increased by 4 inches is \begin{align*}x^2 + 8x+16\end{align*}.

If the height is increased by 4 inches and the width is decreased by 4 inches then we would have the following expression for area:

\begin{align*}A &=(x+4)(x-4)\end{align*}

We can use the Sum and Difference formula,  \begin{align*}(a+b)(a-b) = a^2-b^2\end{align*}, to simplify:

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

The area of the doghouse if the height is increased by 4 inches and the width is decreased by 4 inches is \begin{align*}x^2-16\end{align*}

Example 2

Multiply \begin{align*}(2x+3y)(2x-3y)\end{align*}.

In order to get the hang of the patterns involved in special products, apply the distributive property to see what will happen:

\begin{align*}(2x+3y)(2x-3y)&=\\ 2x(2x-3y)+3y(2x-3y)&=\\ 2x(2x)+2x(-3y)+3y(2x)+3y(-3y)&=\\ 4x^2-6xy+6xy-9y^2&=\\ 4x^2-9y^2\end{align*}

Notice how the two middle terms canceled each other out. This always happens, which is where we get the sum and difference product. Compare the answer above to that from using the sum and difference product:

\begin{align*}(a+b)(a-b) & = a^2-b^2 \\ (2x+3y)(2x-3y) & =(2x)^2-(3y)^2 \\ &=2^2x^2-3^2y^2 \\ & =4x^2-9y^2\end{align*}

The two answers are the same. You can use the sum and difference product as a shortcut, so you don't always have to go through the whole process of multiplying out using the distributive property.

Review

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

  1. \begin{align*}(x+9)^2\end{align*}
  2. \begin{align*}(x-1)^2\end{align*}
  3. \begin{align*}(2y+6)^2\end{align*}
  4. \begin{align*}(3x-7)^2\end{align*}
  5. \begin{align*}(7c+8)^2\end{align*}
  6. \begin{align*}(9a^2+6)^2\end{align*}
  7. \begin{align*}(b^2-1)^2\end{align*}
  8. \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 red 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*}

Review (Answers)

To see the Review answers, open this PDF file and look for section 9.5. 

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Vocabulary

TermDefinition
Square of a Binomial The product of a squared binomial is always a trinomial.
sum and difference formula (a+b)(a-b) = a^2-b^2
Binomial A binomial is an expression with two terms. The prefix 'bi' means 'two'.

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