Binomial multiplication with the distributive property

 
 
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What is the distributive property?

The distributive property can be used even when there are two sets of parentheses with two terms each. It’s called binomial multiplication (remember that a bicycle has two wheels and a binomial has two terms).

Binomial Multiplication:

???(a+b)(c+d)=ac+ad+bc+bd???

???(a-b)(c-d)=ac-ad-bc+bd???

Notice that ???a??? is multiplied by both terms in the second set of parentheses and then ???b??? is multiplied by both terms in the second set of parentheses.

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Using a chart to multiply binomials

We can also make a chart in which the terms ???a??? and ???b??? from the first set of parentheses go across the top, and the terms ???c??? and ???d??? from the second set of parentheses go along the left side. Then we multiply each row by each column to get a result. The four results all get added together to make the expanded polynomial.

 
distributive property with no negatives.png
 

When we add all the results in the chart together, we get

???ac+bc+ad+bd???

When we have negative signs in the binomials, we keep the negative sign with the term that follows it, and our chart looks like

 
distributive property with two negatives.png
 

When we add all the results in the chart together, we get

???ac-bc-ad+bd???

These charts are another way to keep track of the different multiplications that happen during binomial multiplication.

 
 
 

This video walks through multiple examples of how to use the distributive property when you multiply binomials

 

 
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An example of binomial multiplication with the distributive property

Example

Use the distributive property to expand the expression.

???5(x-2)(x+3)???

 

Start by distributing the ???5??? across ???x-2???.

???[5(x)-5(2)](x+3)???

???(5x-10)(x+3)???

Now distribute both of the terms in the first set of parentheses across both of the terms in the second set of parentheses. You may use a chart to help organize your work.

 
distributive property with one negative.png
 

When we add all the results in the chart together, we get

???5x^2+15x-10x-30???

Combine like terms ???15x-10x???.

???5x^2+5x-30???

 
 
distributive property and binomial multiplication

The distributive property can be used even when there are two sets of parentheses with two terms each. It’s called binomial multiplication


Let’s try another example of binomial multiplication.


Example

Use the distributive property to expand the expression.

???3x(x+4)(x+1)(x-2)???

 

Start by distributing the ???3x??? across ???x+4???.

???(3x^2+12x)(x+1)(x-2)???

Now distribute ???3x^2+12x??? across ???x+1???. You may use a chart to help organize your work.

 
distributive property.png
 

When we add all the results in the chart together, we get

???3x^3+3x^2+12x^2+12x???

Combine like terms ???3x^2+12x^2???.

???3x^3+15x^2+12x???

Then distribute ???3x^3+15x^2+12x??? across ???x-2???. You may use a chart to help organize your work.

 
distributive property across three terms.png
 

When we add all the results in the chart together, we get

???3x^4+15x^3+12x^2-6x^3-30x^2-24x???

Combine like terms ???15x^3-6x^3??? and ???12x^2-30x^2???.

???3x^4+9x^3-18x^2-24x???

 
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