x 3+ 5x2
-2x - 24 ÷ x-2
Answers
STEP
1
:
Equation at the end of step 1
((x3) - 5x2) - 24x = 0
STEP
2
:
STEP
3
:
Pulling out like terms
3.1 Pull out like factors :
x3 - 5x2 - 24x = x • (x2 - 5x - 24)
Trying to factor by splitting the middle term
3.2 Factoring x2 - 5x - 24
The first term is, x2 its coefficient is 1 .
The middle term is, -5x its coefficient is -5 .
The last term, "the constant", is -24
Step-1 : Multiply the coefficient of the first term by the constant 1 • -24 = -24
Step-2 : Find two factors of -24 whose sum equals the coefficient of the middle term, which is -5 .
-24 + 1 = -23
-12 + 2 = -10
-8 + 3 = -5 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -8 and 3
x2 - 8x + 3x - 24
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (x-8)
Add up the last 2 terms, pulling out common factors :
3 • (x-8)
Step-5 : Add up the four terms of step 4 :
(x+3) • (x-8)
Which is the desired factorization
Equation at the end of step
3
:
x • (x + 3) • (x - 8) = 0
STEP
4
:
Theory - Roots of a product
4.1 A product of several terms equals zero.
When a product of two or more terms equals zero, then at least one of the terms must be zero.
We shall now solve each term = 0 separately
In other words, we are going to solve as many equations as there are terms in the product
Any solution of term = 0 solves product = 0 as well.