Polynomial Basics: Learn It 3

Multiplying Polynomials

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

  1. Multiply each term of the first polynomial by each term of the second.
  2. Combine like terms.
  3. 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]

Find the product.

[latex]\left(3x+2\right)\left({x}^{3}-4{x}^{2}+7\right)[/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.Two quantities in parentheses are being multiplied, the first being: a times x plus b and the second being: c times x plus d. This expression equals ac times x squared plus ad times x plus bc times x plus bd. The terms ax and cx are labeled: First Terms. The terms ax and d are labeled: Outer Terms. The terms b and cx are labeled: Inner Terms. The terms b and d are labeled: Last Terms.

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

  1. Multiply the first terms of each binomial.
  2. Multiply the outer terms of the binomials.
  3. Multiply the inner terms of the binomials.
  4. Multiply the last terms of each binomial.
  5. Add the products.
  6. 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:


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:

[latex]\left(2x-18\right)\left(3x + 3\right) = 6x^2-48x-54[/latex]

Use FOIL to find the product.[latex]\left(x+7\right)\left(3x - 5\right)[/latex]

definition