Binomio

Earlier, we saw how to multiply binomials using FOIL and using tables.

Distributing Over Polynomials

Any two polynomials can be multiplied with the distributive property. The distributive property states $$a(b+c)=ab+ac$$

Example: Multiplying Binomials

We can use this property to multiply two binomials. Let's look at an example. $$(3x-1)(2x+5)$$

Let's start by multiplying the first term of $(3x-1)$ by the entire second polynomial. $$3x(2x+5)$$

We're going to use the distributive property to expand this. $$3x\cdot 2x + 3x\cdot 5\\ 6x^2 + 15x$$

Now, we do the same process with the second term of the $(3x-1)$. $$-1(2x+5)\\ -1\cdot 2x + -1\cdot 5\\ -2x-5$$

The final step is to combine these two parts and simplify. $$6x^2+15x + (-2x - 5)\\ 6x^2+15x-2x-5\\ 6x^2+13x-5$$

Notice that this is very similar to the FOIL method. This is because FOIL is an acronym to make distribution a little bit easier.

Example: Multiplying Binomials and Other Polynomials

There are cases where the FOIL method doesn't work. These cases require the distributive property. Let's work through an example with a binomial and a trinomial. $$(x+4)(2x^2-x-1)$$

The first term of the binomial needs to be multiplied by the entire trinomial. We'll use the distributive property to expand. $$x(2x^2-x-1)\\ x\cdot 2x^2 + x\cdot (-x) + x \cdot (-1)\\ 2x^3 -x^2 -x$$

Now we do the same process for the second term. $$4(2x^2-x-1)\\ 4\cdot 2x^2 + 4 \cdot (-x) + 4\cdot (-1)\\ 8x^2-4x-4$$

The last step is to combine the two parts and simplify. $$2x^3-x^2-x+(8x^2-4x-4)\\ 2x^3-x^2-x+8x^2-4x-4\\ 2x^3+7x^2-5x-4$$

Practice Problem

We're going to multiply each term of the first polynomial by all of the terms of the second. At the end we'll combine the parts and simplify. $$x^2(3x^2-3x+2)\\ x^2\cdot 3x^2 + x^2 \cdot (-3x) + x^2 \cdot 2\\ 3x^4-3x^3+2x^2\\ \phantom{.}\\ 2x(3x^2-3x+2)\\ 2x\cdot 3x^2 + 2x \cdot (-3x) + 2x \cdot 2\\ 6x^3-6x^2+4x\\ \phantom{.}\\ 1(3x^2-3x+2)\\ 1\cdot 3x^2+ 1\cdot (-3x) + 1\cdot 2\\ 3x^2-3x+2\\ \phantom{.}\\ 3x^4-3x^3+2x^2+6x^3-6x^2+4x+3x^2-3x+2\\ 3x^4-3x^3+6x^3+2x^2-6x^2+3x^2+4x-3x+2\\ 3x^4+3x^3-x^2+x+2$$

You can also go backward and factor out monomials to get back to your additional expression! This is a good way to check your work.

(Video) Multiplying Polynomials Using the Distributive Property

by ProfRobBob

This video by ProfRobBob explains polynomial multiplication with the distributive property.

Summary

The steps he lists to multiply polynomials are:

1. Multiply each term of one polynomial by each term of the other polynomial.
2. Add the resulting monomials (like terms).

Binomial Times Binomial

The best way to understand this is to work through some examples. The first one he tackles is: $$(x+3)(x-1)$$

Start by multiplying the first term in the first polynomial, $x$ by each term in the second polynomial, $x$ and $-1$. $$x\cdot(x-1)\\ x\cdot x + x\cdot -1\\ x^2-x$$

Then, we move to the second term of the first polynomial, 3, and multiply this by each term in the second polynomial, $x$ and $-1$. $$3\cdot(x-1)\\ 3\cdot x + 3\cdot -1\\ 3x-3$$

The final step is to combine the two parts and simplify. $$x^2-x+(3x-3)\\ x^2-x+3x-3\\ x^2+2x-3$$

Let's try one more together. $$(-2a-3)(4a^2+7)$$

First, multiply $-2a$ with $4a^2$ and $7$. $$-2a\cdot (4a^2+7)\\ -2a\cdot 4a^2+ -2a\cdot 7\\ -8a^3-14a$$

Second, multiply $-3$ with $4a^2$ and $7$. $$-3\cdot (4a^2+7)\\ -3\cdot 4a^2+ -3\cdot 7\\ -12a^2-21$$

Finally, write out the terms and simplify. $$-8a^3-14a + (-12a^2-21)\\ -8a^3-14a -12a^2-21\\ -8a^3-12a^2-14a-21$$

Binomial Times Other Polynomials

This method works when multiplying different types of polynomials. Let's look an example with a binomial and a trinomial. This is taken from the video, so you can follow along.

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

First, multiply $x$ with the entire trinomial. $$x\cdot (3x^2+x-4)\\ x\cdot 3x^2+ x\cdot x + x\cdot (-4)\\ 3x^3+x^2-4x$$

Second, multiply $-2$ with the entire trinomial. $$-2\cdot (3x^2+x-4)\\ -2\cdot 3x^2+ -2\cdot x + -2\cdot (-4)\\ -6x^2-2x+8$$

Third, combine these parts and simplify. $$3x^3+x^2-4x + (-6x^2-2x+8)\\ 3x^3+x^2-4x -6x^2-2x+8\\ 3x^3+x^2-6x^2-4x -2x+8\\ 3x^3+-5x^2-6x+8$$

Multiplying Polynomials

We've seen how to use the distributive property to multiply a monomial by a binomial or polynomial. First, we'll look at how to multiply a binomial by a binomial. Then, we'll look at how to multiply a binomial by a trinomial.

Let's dive in!

Multiplying binomials like this is using FOIL, which we have seen before. We multiply each term in one set of parentheses by each term in the second set of parentheses.

Now, let's look at an example where we multiply a binomial by a trinomial.

This second example is a little more tricky, but we can see that we follow a similar pattern as we did when we used FOIL: we multiply each term in the first set of parentheses by each term in the second set of parentheses.

Let's do some practice!

Solution: $$(2x+4)(3x^2+4x+9)$$ Distribute. $$6x^3 +8x^2 +18x+12x^2+16x+36$$ Simplify. $$6x^3 +20x^2 +34x+36$$