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!