Dividing Polynomials by Monomials

Hello! Welcome to this video on dividing polynomials by monomials. Remember, monomials are mathematical expressions with one term, like \(2x^{2}\), and polynomials are mathematical expressions with two or more terms, like \(4y^{2}+3y\). In this video, we will walk through a few examples of how to divide polynomials by monomials.

Let’s start with a simple example.

\(\frac{3x^{2}+4x}{x}\)

Remember, fraction bars always mean divide, so this just means \((3x^{2}+4x)\div x\).

When dividing polynomials by monomials, we want to divide each term of the polynomial by the monomial. Let’s start with \(3x^{2}\). Remember, to divide monomials by one another, you first divide the coefficients. It doesn’t look like there’s a coefficient here \((x)\), but remember, if there’s no coefficient in front of a variable it’s assumed to be 1.

Remember when we have variables, the same variable raised to exponents, you simply subtract the exponents when dividing them. Remember, any variable that’s not raised to a power is really raised to the first power, so this is the same as, \(x^{2}\div x^{1}\), and \(2-1=1\), so we simply have \(x^{1}\). And \(3x^{1}\), of course, can simplify to \(3x\).

\(\frac{3x^{2}}{x}=3x^{1}=3x\)

Now let’s move on to \(4\)x.

\(\frac{4x}{x}=4\cdot 1=4\)

Finally, we simply add the two terms.

\(\frac{3x^{2}+4x}{x}=3x+4\)

Let’s try another example!

\((28y^{5}-14y^{4}-21y^{3}+35y^{2})\div 7y^{2}\)

Remember, we always want to divide each term of the polynomial by the monomial. So let’s start with \(28y^{5}\).

\(\frac{28y^{5}}{7y^{2}}=4y^{3}\)

Now let’s move on to the next term. Signs are very important in math, so always grab the minus sign (or negative sign) if there is one.

\(\frac{-14y^{4}}{7y^{2}}=-2y^{2}\)

Now let’s move on to the next term: \(-21y^{3}\) (remember to grab that negative sign).

\(\frac{-21y^{3}}{7y^{2}}=-3y\)

Finally, we’ll move on to \(35y^{2}\).

\(\frac{35y^{2}}{7y^{2}}=5\)

So that means that our answer is all of these terms added together. Remember adding a negative number is the same as subtracting a number.

\(4y^{3}-2y^{2}-3y+5\)

Before we go, let’s try one that is slightly more challenging.

\(\frac{24y^{6}+16y^{4} -2y^{5} +8y^{2}}{4y3}\)

In this one, not all of the terms are going to divide evenly into whole numbers. Don’t let that intimidate you! Simply follow the same steps we have been, even when things don’t look as pretty. All we have to do is divide each term in the numerator by \(4y^{3}\).

\(\frac{24y^{6}}{4y^{3}}=6y^{3}\)

Now we’ll move on to \(16y^{4}\).

\(\frac{16y^{4}}{4y^{3}}=4y\)

Then we’ll move on to \(-2y^{3}\) (don’t forget to grab the negative sign).

\(\frac{-2y^{3}}{4y^{3}}=-12\)

Then finally, we have \(8^{y}\div 24y^{3}\). Remember, anything to the negative power is in the denominator of a fraction.

\(\frac{8y^{2}}{4y^{3}}=2y\)

Finally, add all the terms together. Our answer is:

\(6y^{3}+4y-\frac{1}{2}+\frac{2}{y}\)

We have fractions in our answer, like I mentioned before. That is totally fine! This is the most simplified form of the answer.

I hope that this video on dividing polynomials by monomials was helpful. Thanks for watching, and happy studying!