Multiplying polynomials is a step up from adding and subtracting them, but once you get the hang of it, it’s pretty straightforward!
To multiply polynomials, we use what’s called the distributive property. This means we take each term from the first polynomial and multiply it by every term in the second polynomial. After that, we just combine any like terms we find.
Distributive Property: [latex]a\cdot(b+c) = a\cdot b+a\cdot c[/latex]How To: Given the multiplication of two polynomials, use the distributive property to simplify the expression
Multiply each term of the first polynomial by each term of the second.
Combine like terms.
Simplify.
Find the product and simplify:[latex]\left(2x+1\right)\left(3{x}^{2}-x+4\right)[/latex]Solution [latex]\begin{align*} (2x+1)(3x^2-x+4) & = 2x(3x^2-x+4) + 1(3x^2-x+4) & \text{Use the distributive property} \\ & = (6x^3-2x^2+8x) + (3x^2-x+4) & \text{Multiply each term} \\ & = 6x^3 + (-2x^2+3x^2) + (8x-x) + 4 & \text{Combine like terms} \\ & = 6x^3 + x^2 + 7x + 4 & \text{Simplify to final form} \end{align*}[/latex]
We can use a table to keep track of our work, as shown in the table below. Write one polynomial across the top and the other down the side. For each box in the table, multiply the term for that row by the term for that column. Then add all of the terms together, combine like terms, and simplify.
We can use a table to keep track of our work as shown below. Write one polynomial across the top and the other down the side. For each box in the table, multiply the term for that row by the term for that column. Then add all of the terms together, combine like terms, and simplify.
[latex]3{x}^{2}[/latex]
[latex]-x[/latex]
[latex]+4[/latex]
[latex]2x[/latex]
[latex]6{x}^{3}\\[/latex]
[latex]-2{x}^{2}[/latex]
[latex]8x[/latex]
[latex]+1[/latex]
[latex]3{x}^{2}[/latex]
[latex]-x[/latex]
[latex]4[/latex]
Using FOIL to Multiply Binomials
For quicker multiplication, especially with binomials, we can use a handy shortcut called the FOIL method.
It is called FOIL because we multiply the First terms, the Outer terms, the Inner terms, and then the Last terms of each binomial.
Visual Example of FOIL with labels
The FOIL method is simply just the distributive property. We are multiplying each term of the first binomial by each term of the second binomial and then combining like terms.
How To: Given two binomials, Multiplying Using FOIL
Multiply the first terms of each binomial.
Multiply the outer terms of the binomials.
Multiply the inner terms of the binomials.
Multiply the last terms of each binomial.
Add the products.
Combine like terms and simplify.
Use the FOIL method to find the product of the polynomials:[latex]\left(2x-18\right)\left(3x + 3\right)[/latex]Solution Find the product of the First terms:
First termsOuter termsInner termsLast terms
Find the product of the Outer terms:
Find the product of the Inner terms:Find the product of the Last terms:
Now combine all the terms obtained from the FOIL method:
[latex]6x^2+6x-54x-54[/latex]
Combine like terms ([latex]6x-54x = -48x[/latex]) and we have found our final simplified product:
