In this unit we will learn how to factor the sum of two cubes. Factor means to un-FOIL a polynomial. The Sum of Two Cubes means that our problems will have two numbers raised to the third power, or cubed.
For example, what polynomials FOILed to make x3 + 1, or x3 + 27? This is the main topic of this unit.
The nice part of factoring two cubes is there is a formula that we can use to do this:
x3 + a3 = (x + a)(x2 - ax + a2)
The left-side of the equals sign represents the standard format of each problem. The right-side of the equals sign is how you would write the answer; replacing variable "a" with its calculated value.
To factor two cubes, you first need to know the value of the cubed numbers.
The first ten cubed numbers are:
13 = 1 x 1 x 1 = 1
23 = 2 x 2 x 2 = 4 x 2 = 8
33 = 3 x 3 x 3 = 9 x 3 = 27
43 = 4 x 4 x 4 = 16 x 4 = 64
53 = 5 x 5 x 5 = 25 x 5 = 125
63 = 216
73 = 343
83 = 512
93 = 729
103 = 1,000
As you work through problems, it will be necessary for you to refer back to this list for each problem. Otherwise you would need to memorize this list.
Factor: x3 + 8
Step 1: Write down the formula:
x3 + a3 = (x + a)(x2 - ax + a2)
Step 2: Re-write 8 as some number cubed. Looking at the list above, we see 23 = 8, so we re-write x3 + 8 as: x3 + 23.
Step 3: We now know on the left-side of the formula that the value of variable "a" is 2. Using substitution into the formula:
x3 + a3 = (x + a)(x2 - ax + a2)
x3 + 23 = (x + 2)(x2 - 2x + 22)
x3 + 23 = (x + 2)(x2 - 2x + 4) *Answer (the right-hand side of the equals sign)
Example: Factor x3 + 64
Step 1: Determine what number cubed is 64 using the list above. 43 = 64, so we re-write 64 as 43
Step 2: Write down the formula:
x3 + a3 = (x + a)(x2 - ax + a2)
Step 3: Substitute 4 for "a":
x3 + 43 = (x + 4)(x2 - 4x + 42)
x3 + 43 = (x + 4)(x2 - 4x + 16) *Answer (the right-hand side of the equals sign)
Example: Factor x3 + 512
Step 1: Determine what number cubed is 512 using the list above. 83 = 512, so we re-write 512 as 83
Step 2: Write down the formula:
x3 + a3 = (x + a)(x2 - ax + a2)
Step 3: Substitute 8 for "a":
x3 + 83 = (x + 8)(x2 - 8x + 82)
x3 + 83 = (x + 8)(x2 - 8x + 64) *Answer (the right-hand side of the equals sign)
Example: Factor x3 + 1000
Step 1: Determine what number cubed is 1,000 using the list above. 103 = 1,000 so we re-write 1,000 as 103
Step 2: Write down the formula:
x3 + a3 = (x + a)(x2 - ax + a2)
Step 3: Substitute 10 for "a":
x3 + 103 = (x + 10)(x2 - 10x + 102)
x3 + 103 = (x + 10)(x2 - 10x + 100) *Answer (the right-hand side of the equals sign)
To check your answer, FOIL the two polynomials together and see if it equals the original problem.
(x + 10)(x2 - 10x + 100)
x3 - 10x2 + 100x + 10x2 - 100x + 1000
x3 - 10x2 + 10x2 + 100x - 100x + 1000
x3 - 0 + 0 + 1000
x3 + 1000 ★
Checking the answer is a bit tedious, but it can be shown we got the correct answer after all the work. That is why having a formula is so nice. It tells you how to write the answer so that you know you got it right, negating the need for checking your answer. But you can always check your answer any time you want to make sure you got it right..
Sum of Two Cubes Calculator
Use this calculator when there is no number infront of x3.
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Sum of Two Cubes Calculator
Use this calculator when there is a number infront of x3.
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