University:
California State University, Northridge
Course:
MATH 102 | Pre-Calculus I
Academic year:

2024
Views:

89

Pages:

2
Author:

dosegamamm9

A binomial is factorable only if it is one of three things a Difference of Squares, a Difference of Cubes, or a Sum of Cubes. A binomial is a Difference of Cubes if both terms are perfect cubes. Recall we may have to factor out a common factor first. If we determine that a binomial is a difference of cubes, we factor it into a binomial and a trinomial. The binomial being the cube root of the first term minus the cube root of the second term. The trinomial comes from the binomial. We square the first term of the binomial, change the sign to addition, multiply the two terms together, and square the second term of the binomial, as in the following formula A3 - B3 = (A - B)(A2 + AB + B2) Factor each of the following. 1. x3 - 27 2. 8x6 - 125 3. 250x4 - 128x #1 solution 1. x3 - 27 We first check that we have a difference of cubes since x3 and 27 are perfect cubes, we do the cube root of x is x and the cube root of 27 is 3 so our binomial is (x-3) to get the first term of the trinomial we square x getting x2 to get the second term of the trinomial we change the sign to + and multiply x by 3, getting +3x to get the third term of the trinomial we square 3 getting 9 so our trinomial is (x2 + 3x + 9) and the answer is (x-3)(x2 + 3x + 9) #2 solution 2. 8x6 - 125 We first check that we have a sum of cubes. since 8x6 and 125 are both perfect cubes we do have a sum of cubes. the cube root of 8x6 is 2x2, and the cube root of 15 is 5 so our binomial is (2x2 - 5) we now use the binomial to create the trinomial we square the first term 2x2 to get 4x4 as our first term we change the sign from - to + and multiply 2x2 and 5 to get +10x2 as our middle term we square the second term 5 to get 25 as our third term so the trinomial is (4x4 + 10x2 + 25) thus our answer is (2x - 5)(4x2 + 10x2 + 25) #3 solution 3. 250x4 - 128x In checking if we have a sum of cubes, we see neither 250x4 or 128x are perfect cubes But we have a common factor we must factor out first, 2x this gives us 2x(125x3 - 64) Now both 125x3 and 64 are perfect cubes the cube root of 125x3 is 5x , and the cube root of 64 is 4 so our binomial is (5x - 4) Now to get the trinomial we square 5x to get 25x2 as our first term we change the sign to + and multiply 5x and 4 to get +20x as the middle term we square 4 to get 16 as our third term so the trinomial is (25x2 + 20x + 16) and the answer is 2x(5x - 4)(25x2 + 20x + 16)