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(Video) Multiplying Polynomials

by mathman1024

This video by mathman1024 covers monomial multiplication.

$\large{\text{Summary}}$

Let's go through the steps for multiplying monomials

1. Multiply the coefficients
2. Use the product rule to multiply variable factors.

Let's try an example:

$$(6x^7)(9x^5)$$

First, we multiply the coefficients.

$$6 \cdot 9 = 54$$

Next, we multiply the factors using the product rule.

$$x^7 \cdot x^5\\ x^{7+5}\\ x^{12}$$

We combine these two to get our answer.

$$(6x^7)(9x^5)=\boxed{54x^{12}}$$

Try these next ones on your own!

1. $(\frac37x^{12}y^8)(\frac54x^{10}y^8)$
2. $\left(\frac{5x^4}{9y^7}\right)^{-2}$

Multiplying Monomials

When multiplying monomials, first multiply the coefficients, then multiply like variables using the exponent product rule.

Let's do an example to practice!

Practice Problem

Solution

$$\begin{align} (2x^6yz^{11})(15x^2z^5)(3y^3)\\ (2 \cdot 15 \cdot 3)(x^6yz^{11})(x^2z^5)(y^3)\\ 90(x^6yz^{11})(x^2z^5)(y^3)\\ 90(x^6 \cdot x^2)(y \cdot y^3)(z^{11} \cdot z^5)\\ 90(x^{6+2})(y^{1+3})(z^{11+5})\\ 90(x^{8})(y^{4})(z^{16})\\ 90x^{8}y^{4}z^{16} \end{align}$$

Example Walk Through

Its daunting to see a lot of variables and numbers! Let's walk through an example and devise a strategy to tackle it.

First, do the following problems to warm up:

$$\begin{align} (2x)(x) &=\\ (2x)(2y) &=\\ (2x)(2xy)&=\\ (2xy^2)(2xy)&= \end{align}$$

Each problem gets a little bit more complicated. Notice how you combine the constants, $x$ and $y$. Now, try the following problem:

I color coded the expression:

When multiplying and dividing monomials, we want to combine like terms—terms with the same base. We combine the constants, the $x$'s and the $y$'s.

Division is similar. Instead of adding the power, we want to subtract (in the right order, of course).

Match the Variables

We usually think of monomials as just one sort of unit in algebra, since mono means one. However, they can be made up of different variables multiplied together!

Let's say we have two monomials that we're trying to multiply: $$(4xy^3z^5) (9x^3z^2)$$ You want to focus on multiplying the like terms of these monomials. The constants multiply together, the $x$ variables multiply together. The $y$ variables multiply together. The $z$ variables multiply together.

The constants, $4$ and $9$, aren't a mystery, so we can multiply them together like we would normally.

The variables are a mystery, so we have to represent how many times they're multiplied with themselves. For multiplying exponents, we add together the exponent parts.

Let's solve the example given above. First, let's split the problem into the multiplication of its like terms. $$\begin{align} &(4xy^3z^5) (9x^3z^2)\\ &=(4\cdot9)\times(x\cdot x^3)\times(y^3)\times(z^5\cdot z^2) \end{align}$$ Then, multiply the like terms. Remember to add the exponent parts of the variables. $$\begin{align} &=36\times x^{1+3}\times y^3\times z^{5+2}\\ &=36x^4y^3z^7 \end{align}$$

Let's try another problem.

Say we want to find the area of a rectangular garden, but we don't know the exact measurements. We do know that the length is equal to $6xyz$ and the width is equal $2xy$.

The area of a rectangle is given by the formula: $$\text{Area}=\text{length}\times\text{width}$$

Multiply the length and the width. When you do this, you multiply the coefficients and add the powers of the variables. $$\begin{align} \text{Area}&=\text{length}\times\text{width}\\ &=(2xy)\times(6xyz)\\ &=(2\cdot6)\times(x\cdot x)\times(y\cdot y)\times(z)\\ &=12\times x^{1+1}\times y^{1\times1}\times z\\ &=12x^2y^2z \end{align}$$