6.4
EQUATIONS CONTAINING PERFECT-SQUARE TRINOMIALS AND DIFFERENCES OF SQUARES
Homework:
Every Other Odd 1-41, All Odd Problems 51-59
| Nancy J. Brien's Home Page | DSPM 0850 Class Notes |
A perfect-square trinomial is the square
of a binomial. It has two identical binomial factors.
A difference of squares is the product
of two binomial conjugates.
Binomial conjugates are binomials that
differ only in the sign of one of their terms.
Factoring Perfect-Square Trinomials:
It is important to be able to recognize a perfect square trinomial. A
perfect-square trinomial has the form 
.
We see that the first and the last terms are squares and must both be
positive. The middle term consists of the product of (
)
multiplied by 2 or by –2.
When factoring a perfect-square trinomial, remember that the binomial factors
are identical.
Example: Factor 
.
By inspection of the trinomial we see that the first and the last terms are
both positive. The first term is the square of x
and the last term is the square of 4. The
middle term is comprised of the product of (x and 4)
multiplied by –2. This is a perfect-square
trinomial and its factors are identical.
Example: Factor 
By inspection, we see that the first and the last terms are both positive. The
first term is the square of 7p and the last
term is the square of 6q. The middle term is
comprised of the product of (7p and 6q)
multiplied by 2.
Note that this factorization could be written as 
.
Factoring a Difference of Squares:
The conjugate of a binomial differs only in the sign of one of its
terms. A+B and A-B are conjugates of each other. When binomial conjugates are
multiplied, the product is a difference of squares. It is important to
recognize a difference of squares. A binomial that is a subtraction of
two terms in which each of the two terms is a square is called a difference of
squares.
Example: Factor 
.
By inspection, we recognize that the binomial is a difference of two squares.
The first term is the square of 2x
and the second term is the square of 7
connected by the operation of subtraction. The
factors are binomial conjugates containing the terms 2x
and 7 in that order.
Example: Factor 
.
We recognize that the binomial is a difference of two squares. The first term
is the square of 
and
the second term is the square of 
connected
by the operation of subtraction. Its factors
are binomial conjugates containing the terms 
and
in
that order.
Using Factoring To Solve Equations:
An equation containing a perfect-square trinomial is said to have a double
root since the factors of a perfect-square trinomial are the same.
Example:
Solve the equation 
.
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Zero is on one side of the equation and we factor the trinomial on the other side. |
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Since the factor is repeated, we need only to set one factor equal to zero and solve. We say that we have a double root. |
An equation consisting of a difference of squares will always have two
solutions that are opposites.
Example:
Solve the equation 
.
|
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Zero is on one side of the equation and we factor the difference of squares on the other side. |
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Setting
both factors equal to zero and solving, we see that the solutions are
opposites, |
.