Quadratic Expressions
Factoring
All, or some the terms in the expression have a GCF.
The expression has two identical binomials in it.
ANSWER: To factor a expression with two identical binomials, think of the binomial as a single term. The GCF will always be the binomial. Divide each term by the binomial to factor the expression.
EXAMPLE: a(b+1) + 9c(b + 1)
Think of (b+1) as a single term. The GCF is the binomial (b+1). Factor out this binomial to get the answer:
(b+1)(a + 9c)
The polynomial doesn't have a common factor for all terms,
but some terms have a common factor.
bu
ANSWER: Here, you have to factor by grouping.
To do this, factor groups of two terms with a common
factor to produce binomials. Then, use the binomial common
factor technique.
EXAMPLE: mx + my + 2x + 2y
Group mx + my into one group, and 2x = 2y in another. Then,
find the GCFs of each group and factor accordingly. You should
get the following:
m(x + y) + 2(x+ y)
Use binomial common factor to factor out (x+y).
(x + y)(m + 2)
The expression is made up of monomials.
ANSWER: First, find the GCF of all the terns. Then,
simply factor out the GCF to get your answer.
EXAMPLE: 15w + 25z
The GCF of 15 and 25 is 5. Divide both terms by 5 to get
the following:
5(3w + 5z)
NOTE: For polynomials with more than one variable, the GCF of the variable is the product of the common bases with the lowest exponent.
None of the terms in the expression have a GCF.
The expression is a trinomial.
The trinomial is a perfect square trinomial.
(NOTE: Verify that the trinomial is a perfect
square by checking if a and c are perfect squares.
Then, multiply both by 2, and if they are equal to
b, you have a perfect square trinomial).
ANSWER: First, verify the the
trinomial is a perfect square trinomial.
Secondly, factor the trinomial as
a2 + 2ab + b2 = (a + b)2.
EXAMPLE: x2 + 20x + 100
Take a look at the terms x2 and 100.
They are both perfect squares.
√x2 + x
√100 = 10
Now check if 2(10)(x) is equal to 20x.
2(10)(x) = 20x
Now that we know this is a perfect square trionmial,
we can factor it as a2 + 2ab + b2 = (a + b)2.
x2 + 20x + 100 = (x)2 + 2(x)(10) + (10)2
= (x + 10)2
The trinomial is NOT a perfect square trinomial.
The trinomial is in the form of x2 + bx + c (a=1).
ANSWER: Find two integers whose product is c and
whose sum is b. Then, express the quadratic expression
as a product.
EXAMPLE: x2 + 7x + 10
The two integers whose product is 10 and sum is 7 are
2 and 5. Now, take the integers and express them as a
product.
x2 + 7x + 10 = (x + 2)(x+5)
The trinomial is in the form of ax2 + bx + c (a≠1)
ANSWER: Find two integers whose product is
a x c and whose sum is b. Then, break up the terms
into groups and factor by grouping.
EXAMPLE: 4x2+ 28xy + 49y2
The two integers whose product is (4)(49) and sum is
(28) are 14 and 14. Now, break up the second term.
4x2 + 14xy + 14xy + 49y2
Factor by grouping to get your answer:
The expression is not a trinomial.
ANSWER: The expression cannot be factored.
Expanding
The expression is a perfect square.
The expression is in the form of (a + b)2
ANSWER: If you are able to recognize the pattern for
squaring a binomial, you can expand it to the format of
a2 + 2ab + b2.
EXAMPLE:
(m + 11)2
Use the appropriate pattern for squaring a binomial to
expand.
(m+11)2 = (m)2 + 2(m)(11) + (11)2
Simplify to get your answer.
m2 + 22m + 121
The expression is in the form of a2 - b2
ANSWER: Here we are dealing with a difference of squares.
When you multiply the sum and difference of two terms, the two middle terms are opposite and add to zero. This is another special product pattern.
EXAMPLE: (v+1)(v-1)
Use the pattern for the product of a sum and difference .
(v+1)(v-1) = (v)2-(v)2
= v2-v2
The expression is not a perfect square.
The expression is made up of two binomials.
ANSWER: To expand this expression,
you have to multiply the two binomials
using the FOIL technique:
Firsts
Outers
Inners
Lasts
EXAMPLE: (x + 3)(x + 5)
After using the FOIL technique, you should
come to the following terms:
x2 + 5x + 3x + 15
Now, add like terms to get your answer:
x2 + 8x + 15