Simplify - 1)7x^2+18x+18/49x^2-16 x 14x-8/x+2
2)z^3 -9y^2 / z^3 -27^3
Answers
STEP 1
:
14x - 8
Simplify ———————
x + 2
STEP
2
:
Pulling out like terms
2.1 Pull out like factors :
14x - 8 = 2 • (7x - 4)
Equation at the end of step
2
:
(((7•(x2))+18x)+8) 2•(7x-4)
——————————————————•————————
((49•(x2))-16) x+2
STEP
3
:
Equation at the end of step
3
:
(((7•(x2))+18x)+8) 2•(7x-4)
——————————————————•————————
(72x2-16) x+2
STEP
4
:
Equation at the end of step
4
:
((7x2+18x)+8) 2•(7x-4)
—————————————•————————
(49x2-16) x+2
STEP
5
:
7x2 + 18x + 8
Simplify —————————————
49x2 - 16
Trying to factor by splitting the middle term
5.1 Factoring 7x2 + 18x + 8
The first term is, 7x2 its coefficient is 7 .
The middle term is, +18x its coefficient is 18 .
The last term, "the constant", is +8
Step-1 : Multiply the coefficient of the first term by the constant 7 • 8 = 56
Step-2 : Find two factors of 56 whose sum equals the coefficient of the middle term, which is 18 .
-56 + -1 = -57
-28 + -2 = -30
-14 + -4 = -18
-8 + -7 = -15
-7 + -8 = -15
-4 + -14 = -18
-2 + -28 = -30
-1 + -56 = -57
1 + 56 = 57
2 + 28 = 30
4 + 14 = 18 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, 4 and 14
7x2 + 4x + 14x + 8
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (7x+4)
Add up the last 2 terms, pulling out common factors :
2 • (7x+4)
Step-5 : Add up the four terms of step 4 :
(x+2) • (7x+4)
Which is the desired factorization
Trying to factor as a Difference of Squares:
5.2 Factoring: 49x2-16
Theory : A difference of two perfect squares, A2 - B2 can be factored into (A+B) • (A-B)
Proof : (A+B) • (A-B) =
A2 - AB + BA - B2 =
A2 - AB + AB - B2 =
A2 - B2
Note : AB = BA is the commutative property of multiplication.
Note : - AB + AB equals zero and is therefore eliminated from the expression.
Check : 49 is the square of 7
Check : 16 is the square of 4
Check : x2 is the square of x1
Factorization is : (7x + 4) • (7x - 4)
Canceling Out :
5.3 Cancel out (7x + 4) which appears on both sides of the fraction line.
Equation at the end of step
5
:
(x + 2) 2 • (7x - 4)
——————— • ————————————
7x - 4 x + 2
STEP
6
:
Canceling Out
6.1 Cancel out (x+2) which appears on both sides of the fraction line.
Canceling Out :
6.2 Cancel out (7x-4) which appears on both sides of the fraction line.
Final result :
2