Factoring Sum and Difference of Cubes #1
Description/Explanation/Highlights
Video Description
In this video we explore factoring the sum and difference of cubes.
Steps and Key Points to Remember
To factor the sum or difference of two perfect cubes with a lead coefficient of 1, follow these steps:
- We will focus on the sum of cubes first but the steps will be the same for the difference with only the signs changing.
- Before starting, make sure all numbers and variables in both terms are perfect cubes. In the example \(x^3+27\) all of the numbers and variables in both terms of the expression are perfect cubes. i.e. \(x^3 = x \cdot x \cdot x\) and \(27= 3 \cdot 3 \cdot 3\)
- The sum or difference of cubes will factor into a binomial (two terms) and a trinomial (three terms) with two sets of parenthesis.
i.e \(\text{(}\hspace{2em}\text{)} \text{(} \hspace{3em} \text{)}\) - At this point, find the cube root of the first term. In other words, what number or variable multiplied by itself three times will give you the first term. In our example \(\sqrt[3]{x^3}=x\)
- Therefore, x is the first term of the binomial.
\(\text{(}x\hspace{1em}\text{)} \text{(} \hspace{3em} \text{)}\) - Repeat the process and find the cube root of the second term. In our example \(\sqrt[3]{27}=3\)
- 3 is the second term of the binomial.
i.e. \(\text{(}x\hspace{1em}\text{3)} \text{(} \hspace{3em} \text{)}\) - Use the same sign as the original problem between the binomial terms. In this example use “+” i.e. \((x+3)\text{( )}\)
- We will find the three terms of the trinomial by using the binomial factor we just found in the first steps of the problem.
- Square the first term of the binomial we just found. This will be the first term of the trinomial. In our example the first term was x, so we should square x and put the result as the first term of the trinomial i.e. \((x+3)\text{(}x^2\hspace{2em}\text{)}\)
- Use the opposite sign from the binomial next. In this case the sign of the binomial is “+” so we should use “-” as the first sign in the trinomial i.e. \((x+3)\text{(}x^2-\hspace{2em}\text{)}\)
- Next, multiply the first and second terms of the binomial together, ignoring the signs. The result will be the second term of the trinomial i.e. \((x+3)\text{(}x^2-3x\hspace{1.5em}\text{)}\)
- The second sign of the trinomial is always positive. i.e. \((x+3)\text{(}x^2-3x+\hspace{1em}\text{)}\)
- Finally, to get the last term of the trinomial, square the second term of the binomial. In our example, square the 3 to get 9. i.e. \((x+3)\text{(}x^2-3x+9\text{)}\)
- We would do exactly the same thing to factor \((x^3-27)\) except the sign of the binomial would be “-” (remember, same as the original) and the first sign of the trinomial would now be “+” (opposite of the binomial sign). The last sign would still be “+” (always). The factored version would then be \((x-3)\text{(}x^2+3x+9\text{)}\)
Here are some key points to keep in mind when factoring the sum or difference in two perfect cubes:
- The rules are the same for sum and difference of cubes except for the signs.
- Both terms must be perfect cubes including all coefficients and variables.
- The sum or difference of cubes will always factor to a binomial and a trinomial.
- The binomial factor is always the cube root of the first term of the original problem separated from the cube root of the second term of the original problem by the same sign as the original problem. So, sums will always have a “+” sign and differences a “-” sign.
- The trinomial factor is derived from the binomial factor not the original problem.
- The first trinomial term is the square of the first term in the binomial factor.
- The first sign in the trinomial factor is always opposite of the sign in the binomial factor.
- The second term of the trinomial factor is the terms of the binomial factor multiplied together without considering the sign.
- The last sign of the trinomial is always positive.
- The last term of the trinomial is the last term of the binomial factor squared.
- It may be helpful to use the acronym SOAP to determine the three signs of the factors: Same sign as the original, Opposite sign of the binomial factor, Always Positive for the last sign.
Video Highlights
- 00:00 \(x^3+27\) example of factoring the sum of cubes
- 02:15 \(x^3-27\) example of factoring the difference of cubes.
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