Multiplication of polynomials
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.
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-
3xy * x2 = 3x3y
Multiply the outside terms of each binomial. (O)
3xy * 2xy2 = 6x2y3
Multiply the inside terms of each binomial. (I) 2x * x2 = 2x3
Multiply the last terms of each binomial. (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.)
There is a special formula for this situation, so you don't have to factor the binomial. The difference of squares formula is listed below.
A2 - B2 = (A + B)(A - B)
2. Problem: Factor y2 - 4.
Solution: Since y2 is the square
of y, and 4 is...