Multiplying Polynomials by Monomials

Hello! Today we’re going to take a look at how to multiply a polynomial by a monomial. Let’s start with a simple example.

$$3x(4x+2)$$

When multiplying a polynomial by a monomial, you’ll want to multiply each part of the polynomial $$(4x+2)$$ by each part of the monomial $$(3x)$$. You’ll multiply the monomial, $$3x$$, by the first part of the polynomial, so by this $$4x$$. And then we’ll add — multiplying the monomial by this part of the polynomial, the $$+2$$.

$$(3x)(4x)+(3x)(2)$$

So now all we have to do is multiply each of these sets of monomials.

$$12x^{2}+6x$$

And that’s our answer. Not too challenging! Let’s try another one.

$$7y^{2}(2y^{2}+6y-9)$$

Remember, multiply each part of the polynomial by the monomial.

$$(7y^{2})(2y^{2})+(7y^{2})(6y)+(7y^{2})(-9)$$

Notice that I put plus signs in between each set of multiplied terms, even though there is a –9. I did this because subtraction is always the same thing as adding a negative number. Our last sign will end up turning into a subtraction symbol, but doing it this way helps us make sure we have the correct sign attached to each term.

So now let’s multiply all of our terms. $$7y^{2}\cdot 2y^{2}=14y^{4}$$. Remember, when you multiply exponential terms with the same base you simply add the exponents, so we’ve got $$2+2=4$$ — so that’s where the $$y^{4}$$ came from. And then, $$7y^{2}\cdot 6y=42y^{3}$$, so +42y3. And then, plus $$7y^{2}\cdot (-9)=-63y^{2}$$.

$$14y^{4}+42y^{3}+(-63y^{2})$$

Remember what I said earlier? Adding a negative number is the same as subtracting, so we can rewrite this as:

$$14y^{4}+42y^{3}-63y^{2}$$

Let’s try one last problem before we go.

$$-5x^{2}y(3x-5xy+10y^{2})$$

We need to multiply each term of the polynomial by the monomial.

$$(-5x^{2}y)(3x)+(-5x^{2}y)(-5xy)+(-5x^{2}y)(10y^{2})$$

Now we multiply each set of monomials. Remember, a negative number times a negative number is a positive number.

$$-15x^{3}y+25x^{3}y^{2}+(-50x^{2}y^{3})$$

We can simplify this last sign, and our final answer will be:

$$-15x^{3}y+25x^{3}y^{2}-50x^{2}y^{3}$$

I hope that this video on multiplying polynomials by monomials has been helpful! Thanks for watching, and happy studying!

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by Mometrix Test Preparation | This Page Last Updated: May 13, 2022