Factor polynomials
Objectives
Student will be able to:
factor the difference of two squares.
factor perfect square trinomials.
factor polynomials by finding the greatest common factor.
Introduction
To
factor
a polynomial means to rewrite it as a product. First, determine if there is a greatest common factor (GCF). If so, factor out the GCF. Then, determine if the resulting polynomial can be factored using the difference of squares pattern or perfect square trinomial pattern.
Factor this polynomial by factoring out the GCF.
First, find the greatest common factor of all the terms. The coefficients, 5 and 20, have a common factor of 5.
Next, look at the variables. Each term includes the variables
x
and
y
. Determine the greatest number of
x
and
y
variables each term has in common.
The first term can be rewritten as
The second term can be rewritten as
.
Both terms have 5,
, and
y
in common. So, the greatest common factor is
.
Factor the GCF from the polynomial.
After the greatest common factor has been factored, look to see if the remaining polynomial can be factored. In this case, it cannot. Remember: to check if factoring is correct, multiply.
A polynomial can often be factored using the difference of two squares pattern or the perfect square trinomial pattern.
The difference of two squares pattern can be applied to a binomial in which one perfect square is subtracted from another perfect square. When factored, the resulting binomials are almost the same, but one binomial is a sum and the other is a difference. The first term in each binomial is the square root of the first perfect square. The second term in each binomial is the square root of the subtracted perfect square.
Example:
Multiply to check the factoring.
Note: This pattern can be applied only to a
difference
of two squares. If the square terms are a sum, the binomial cannot be factored. For example,
cannot be factored.
Polynomials can also be factored using the perfect square trinomial pattern. Look at the examples below. Notice that the first and last terms of each trinomial are perfect squares. Also, the last term of each trinomial is positive. When factored, a perfect square trinomial produces a squared binomial.
Example:
Multiply to check the factoring.
Example:
Multiply to check the factoring.
The techniques for factoring shown above can be combined to factor many polynomials. Follow these steps:
Look for a greatest common factor and factor it out.
Look at what remains in the parentheses and determine if it can be factored. If it can be factored, factor using one of the known patterns.
Factor the binomial.
Determine if there is a GCF. In this expression, both terms have a common factor of 2.
Factor the 2.
Next, check to see if the result can be factored. Notice it is the difference of two squares (
and 16), so use the difference of two squares pattern.
To factor, determine the square root of
and the square root of 16.
Since
and
,the values for
a
and
b
are
x
and 4.
Factor the polynomial.
Since there is no factor besdes 1 common to all three terms, look to see if there is a pattern. Notice the first term,
, and the last term, 36, are perfect squares. This polynomial is a perfect square trinomial.
To factor, determine the square root of
and the square root of 36.
Since
and
, the values for
a
and
b
are
x
and 6.
Examples
1.
Factor the polynomial.
Factor the GCF from the polynomial.
The GCF of the numbers is 3. There is no common variable in all three terms since 48 does not have an
x
. So, factor 3 from the polynomial.
Now, check to see if the resulting polynomial can be factored by using a special pattern.
The resulting polynomial inside the parentheses is a perfect square trinomial because
and 16 are both perfect squares. Factor the perfect square trinomial.
2.
Factor the binomial.
Factor out the GCF. The numbers 18 and 12 have a common factor of 6. Each variable term has at least an
The GCF is
Factor
Always check to see if the resulting expression can be factored again. In this case, the expression inside the parentheses cannot be factored.
Try Together
Complete the exercises. Show all work.
1.
Factor the binomial.
Check student's work.
2.
Factor the polynomial.
Check student's work.
3.
Factor the binomial.
(Hint: First factor out a GCF, then factor what is left inside the parentheses.)
Check student's work.
Please evaluate the student's answers:
Excellent
Good
Okay