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On this page we hope to clear up problems you might have
with polynomials and factoring. All the different methods
of factoring and different things such as the difference
of cubes are covered. Click any of the links below or scroll
down to start gaining a better understanding of polynomials
and factoring.
Multiplication of polynomials Factoring Factoring by grouping Sums and differences of cubes Quiz on Polynomials and Factoring
When terms of a polynomial have the same variables raised to the same powers, the terms are called similar, or like terms. Like terms can be combined to make the polynomial easier to deal with. Example: 1. Problem: Combine like terms in the following
equation: 3x2 - 4y + 2x2.
Solution: Rearrange the terms so it is easier
to deal with.
3x2 + 2x2 - 4y
Combine the like terms.
5x2 - 4y
Probably the most important kind of polynomial multiplication that you can learn is the multiplication of binomials (polynomials with two terms). An easy way to remember how to multiply binomials is the FOIL method, which stands for first, outside, inside, last. Example:
1. Problem: Multiply (3xy + 2x)(x^2 + 2xy^2).
Simplify the answer.
Solution: Multiply the first terms of each bi-
nomial. (F)
3xy * x2 = 3x3y
Multiply the outside terms of each bi-
nomial. (O)
3xy * 2xy2 = 6x2y3
Multiply the inside terms of each bi-
nomial. (I)
2x * x2 = 2x3
Multiply the last terms of each bi-
nomial. (L)
2x * 2xy2 = 4x2y2
You now have a polynomial with four terms.
Combine like terms if you can
to get a simplified answer.
There are no like terms, so you have your final answer.
3x3y + 6x2y3 + 2x3 + 4x2y2
Although it would be nice if all you ever had to do was multiply binomials by other binomials, that isn't even close to reality. A perfect example of this is when you have to cube a binomial. Example: 2. Problem: Multiply (A + B)3 out.
Solution: Rewrite so you have something you can
actually multiply out.
(A + B)(A + B)(A + B)
Multiply the first two binomials together.
(A + B)(A + B)
A2 + AB + BA + B2
After combining like terms, you have
A2 + 2AB + B2
You now have a binomial and a
trinomial to multiply together.
(A2 + 2AB + B2)(A + B)
This is a slightly more complicated
situation than multiplying a binomial
by another binomial. Multiply the
first term of the binomial by each of
the terms in the trinomial and then
multiply the last term of the binomial
by each term in the trinomial.
A3 + 2A2B + AB2 + BA2 + 2AB2 + B3
Combine like terms if possible to
simplify the answer.
A3 + 3A2B + 3AB2 + B3
Factoring is the reverse of multiplication. When factoring, look for common factors. Example: 1. Problem: Factor out of a common factor of
4y2 - 8.
Solution: 4 is a common factor of
both terms, so pull it out and write
each term as a product of factors.
4y2 - (4)2
Rewrite using the distributive law of
multiplication, which says that
a(b + c) = ab + ac.
4(y2 - 2)
Sometimes, you will come across a special situation where both terms
of a binomial are squares of another number, such as (x2 + 9).
(x2 is the square of x and 9 is the
square of 3.)
2. Problem: Factor y2 - 4.
Solution: Since y2 is the square
of y, and 4 is the square of 2,
this binomial fits the difference
of squares formula.
y2 - 4 = (y + 2)(y - 2)
Since trinomials are the most common polynomial you will be asked
to factor, we will try to help you better understand how to
factor quadratic trinomials, or trinomials whose highest power is
two. Also, we assume you know how to multiply
binomials (we use the "FOIL" method).
3. Problem: Factor: x2 - 14x - 15.
Solution: First, write down two sets of parentheses to indicate the
product.
( )( )
Since the first term in the trinomial is the product of the
first terms of the binomials, you enter x as the first
term of each binomial.
(x )(x )
The product of the last terms of the binomials must equal
-15, and their sum must equal -14, and one of the
binomials' terms has to be negative. Four different pairs of
factors have a product that equals -15.
(3)(-5) = -15 (-15)(1) = -15
(-3)(5) = -15 (15)(-1) = -15
However, only one of those pairs has a sum of -14.
(-15) + (1) = -14
Therefore, the second terms in the binomial are -15 and
1 because these are the only two factors whose product
is -15 (the last term of the trinomial) and whose sum
is -14 (the coefficient of the middle term in
the trinomial).
(x - 15)(x + 1) is the answer.
Trinomials and binomials are the most common polynomials, but you will sometimes see polynomials with more than three terms. Sometimes, when you are dealing with polynomials with four or more terms, you can group the terms in such a way that common factors can be found. Example:
4. Problem: Factor 4x2 - 3x + 20x - 15.
Solution: Rearrange the terms so common
factors can be more easily found.
4x2 + 20x - 3x - 15
The first two terms have a common factor
in 4x. The last two terms have a
common factor in 3. Factor those
terms out.
4x(x + 5) - 3(x + 5)
Now you have a binomial. Each term
has a factor of (x + 5). Factor
that out for the final answer.
(x + 5)(4x - 3)
When you have a binomial where each term is the cube of something, such as x3 - 27, there are
formulas that you can apply to the binomial so you don't have to factor them. The two formulas are listed
below.
1. Problem: Factor 125x3 + y3.
Solution: Write the sum of the cube roots.
(5x + y)( )
Take 5x + y to get the next factor.
Think of the Sum of Cubes formula
(A + B)(A2 - AB + B2).
5x would be A and y would
be B.
(5x + y)(25x2 - 5xy + y2)
Take the Quiz on polynomials and factoring. (Very useful to review or to see if you've really got this topic down.) Do it! |
