Section 7.4 Special Products and Factors
Subsection Squares of Monomials
A few special binomial products occur so frequently that it is useful to recognize their forms. This will enable you to write their factored forms directly, without trial and error. To prepare for these special products, we first consider the squares of monomials.
Study the squares of monomials in Example 7.30. Do you see a quick way to find the product?
Look Closer.
In Example 7.30a, we doubled the exponent and kept the same base. In Example 7.30b, we squared the numerical coefficient and doubled the exponent.
Reading Questions Reading Questions
1.
Why do we double the exponent when we square a power?
Answer.
Because when we multiply powers with the same base, we add the exponents.
Example 7.31.
Find a monomial whose square is
Solution.
When we square a power, we double the exponent, so is the square of Because 36 is the square of 6, the monomial we want is To check our result, we square to see that
Subsection Squares of Binomials
Squares of Binomials.
Reading Questions Reading Questions
2.
Explain why it is NOT true that
Answer.
Because we must square using FOIL.
We can use these results as formulas to compute the square of any binomial.
Example 7.32.
Expand as a polynomial.
Solution.
The formula for the square of a sum says to square the first term, add twice the product of the two terms, then add the square of the second term. We replace by and by in the formula.
Of course, you can verify that you will get the same answer for Example 7.32 if you compute the square by multiplying
Caution 7.33.
Reading Questions Reading Questions
3.
How do we compute
Answer.
Subsection Difference of Two Squares
Now consider the product
In this product, the two middle terms cancel each other, and we are left with a difference of two squares.
Difference of Two Squares.
Example 7.34.
Multiply
Solution.
The product has the form with replaced by and replaced by We use the difference of squares formula to write the product as a polynomial.
Reading Questions Reading Questions
4.
Answer.
To simplify we must square the binomial.
Subsection Factoring Special Products
The three special products we have just studied are useful as patterns for factoring certain polynomials. For factoring, we view the formulas from right to left.
Special Factorizations.
If we recognize one of the special forms, we can use the formula to factor it. Notice that all three special products involve two squared terms, and so we first look for two squared terms in our trinomial.
Example 7.35.
Factor
Solution.
This trinomial has two squared terms, and These terms are and so and We check whether the middle term is equal to
This is the correct middle term, so our trinomial has the form (1), with and Thus,
Reading Questions Reading Questions
5.
How can we factor
Answer.
6.
How can we factor
Answer.
Caution 7.36.
Sum of Two Squares.
The sum of two squares, cannot be factored.
As always when factoring, we should check first for common factors.
Example 7.37.
Factor completely
Solution.
Each term has a factor of 2, so we begin by factoring out 2.
The polynomial in parentheses has the form with and The middle term is
We use equation (2) to write
Thus,
Reading Questions Reading Questions
7.
What expression involving squares cannot be factored?
Answer.
Subsection Skills Warm-Up
Exercises Exercises
Subsubsection Answers to Skills Warm-Up
Subsection Lesson
Subsubsection Activity 1: Special Products
Exercises Exercises
1.
2.
3.
Subsubsection Activity 2: Special Factorizations
Exercises Exercises
1.
2.
Subsubsection Wrap-Up
Objectives.
Questions.
- Explain the difference between squaring
and squaring - How can you check whether a trinomial might be the square of a binomial?
- Explain why we can factor the difference of two squares, but we cannot factor the sum of two squares.
Subsection Homework Preview
Exercises Exercises
Subsubsection Answers to Homework Preview
Exercises Homework 7.4
1.
2.
Exercise Group.
Exercise Group.
For Problems 14–19, use the formula for difference of two squares to multiply.
Exercise Group.
For Problems 32–40, factor completely.
41.
42.
Use areas to explain why the figure illustrates the product
43.
44.
45.
- Expand
by multiplying. - Use your formula from part (a) to expand
- Substitute
and to show that is not equivalent to
46.
- Multiply
- Factor
- Factor