Final result :
(x + 5) • (x + 4)
———————
2
Step by step solution :
Step 1 :
x2 - 16
Simplify ———————
2x + 6
Step 2 :
Pulling out like terms :
2.1 Pull out like factors :
2x + 6 = 2 • (x + 3)
Trying to factor as a Difference of Squares :
2.2 Factoring: x2 - 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 : 16 is the square of 4
Check : x2 is the square of x1
Factorization is : (x + 4) • (x - 4)
Polynomial Long Division :
2.3 Polynomial Long Division
Dividing : x + 4
("Dividend")
By : x + 3 ("Divisor")
dividend x + 4
- divisor * x0 x + 3
remainder 1
Quotient : 1
Remainder : 1
Equation at the end of step 2 :
(((x2)+8x)+15) (x+4)•(x-4)
——————————————•———————————
(x-4) 2•(x+3)
Step 3 :
x2 + 8x + 15
Simplify ————————————
x - 4
Trying to factor by splitting the middle term
3.1 Factoring x2 + 8x + 15
The first term is, x2 its coefficient is 1 .
The middle term is, +8x its coefficient is 8 .
The last term, "the constant", is +15
Step-1 : Multiply the coefficient of the first term by the constant 1 • 15 = 15
Step-2 : Find two factors of 15 whose sum equals the coefficient of the middle term, which is 8 .
-15 + -1 = -16
-5 + -3 = -8
-3 + -5 = -8
-1 + -15 = -16
1 + 15 = 16
3 + 5 = 8 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, 3 and 5
x2 + 3x + 5x + 15
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (x+3)
Add up the last 2 terms, pulling out common factors :
5 • (x+3)
Step-5 : Add up the four terms of step 4 :
(x+5) • (x+3)
Which is the desired factorization
Equation at the end of step 3 :
(x + 5) • (x + 3) (x + 4) • (x - 4)
————————————————— • —————————————————
x - 4 2 • (x + 3)
Step 4 :
Canceling Out :
4.1 Cancel out (x+3) which appears on both sides of the fraction line.
Canceling Out :
4.2 Cancel out (x-4) which appears on both sides of the fraction line.
Final result :
(x + 5) • (x + 4)
——————---
2
