PE - Multiply Polynomials Lesson

Multiply Polynomials

When you multiply two like expressions keep the base and add the exponents.

Examples

1. $LaTeX: x^4\cdot x^2=x^6$ $x^4\cdot x^2=x^6$

2. $LaTeX: 4b^2\cdot2b^7=8b^9$ $4b^2\cdot2b^7=8b^9$

Multiplying Monomials Practice

Try these problems!

$LaTeX: \left(3x^2\right)\left(4x^4\right)$ $\left(3x^2\right)\left(4x^4\right)$
$LaTeX: \left(4x^4\right)\left(5x^5\right)$ $\left(4x^4\right)\left(5x^5\right)$
$LaTeX: \left(x^2y^2\right)\left(x^3y\right)$ $\left(x^2y^2\right)\left(x^3y\right)$
$LaTeX: \left(-2x^2y\right)\left(3x^8y^4\right)$ $\left(-2x^2y\right)\left(3x^8y^4\right)$

TO VIEW THE SOLUTIONS ONCE YOU HAVE PRACTICED, CLICK HERE. Links to an external site.

The distributive property is used to multiply monomials and polynomials. Here is an example:

You try:

1. $LaTeX: \langle3x\rangle\langle4x^2+2x-1\rangle$ $\langle3x\rangle\langle4x^2+2x-1\rangle$

2. $LaTeX: \langle xy^2\rangle\langle2x+3y-4\rangle$ $\langle xy^2\rangle\langle2x+3y-4\rangle$

There are times when multiplying a binomial by a binomial is very expedient. This process is referred to as the FOIL method which is the same as "distributing twice." This acronym stands for the terms of of the product.

FOIL METHOD: multiple First, Outer, Inner, Last
FIRST: (x+3)(x+2) with the x bolded
OUTER: (x+3)(x+2) with first x and 2 bolded
INNER: (x+3)(x+2) with the 3 and second x bolded
LAST: (x+3)(x+2) with the 3 and 2 bolded
x squared + 2x+3x+6
x squared+5x+6

Watch this video for more examples of multiplying polynomials. The examples include:

Multiplying a monomial and polynomial.
Multiplying two binomials using the FOIL method.
Multiplying a binomial and a trinomial.
Word problem finding the area.

A common mistake in algebra is to distribute an exponent over an addition or subtraction symbol. Be sure you always write the binomials twice and FOIL

_______________________________________________________________________

Binomial Theorem

What if we wanted to multiply a binomial by itself more than twice? Multiplying equation image indicator $LaTeX: \left(w-3\right)^3=\left(w-3\right)\left(w-3\right)\left(w-3\right)$ $\left(w-3\right)^3=\left(w-3\right)\left(w-3\right)\left(w-3\right)$ could take a long time, but it can be done "fairly quickly". What if you wanted to multiply this? $LaTeX: \left(w-3\right)^{11}$ $\left(w-3\right)^{11}$ To expand this expression, we would have to multiply (w-3) to itself 11 times. This would be very tedious and leave a lot of room for error.

A faster way to expand this expression is to use the Binomial Theorem, which tells us that we can use Pascal's Triangle to find the coefficients for the terms. What is Pascal's triangle?

Binomial Coefficients in Pascal's Triangle

Pascal's triangle can be used to determine the coefficients in binomial expansions.

For example:

equation image indicator $LaTeX: \left(x+y\right)^0=1$ $\left(x+y\right)^0=1$

$LaTeX: \left(x+y\right)^1=x+y=1x^1y^0+1x^0y^1$ $\left(x+y\right)^1=x+y=1x^1y^0+1x^0y^1$

$LaTeX: \left(x+y\right)^2=x^2+2xy+y^2=1x^2y+2x^1y^1+1x^0y^2$ $\left(x+y\right)^2=x^2+2xy+y^2=1x^2y+2x^1y^1+1x^0y^2$

$LaTeX: \left(x+y\right)^3=x^3+3x^2y+3xy^2+y^3=1x^3y^0+3x^2y^1+3x^1y^2+1x^0y^3$ $\left(x+y\right)^3=x^3+3x^2y+3xy^2+y^3=1x^3y^0+3x^2y^1+3x^1y^2+1x^0y^3$

Notice that the coefficients of equation image indicator $LaTeX: \left(x+y\right)^n$ $\left(x+y\right)^n$ correspond to numbers in row n of Pascal's Triangle.

Multiplying Polynomials Self-Assessment Activity

IMAGES CREATED BY GAVS