(4x²-16x+15)÷(2x-3)
Factorise and divide.
Answers
STEP
1
:
Equation at the end of step 1
((0 - 22x2) - 16x) - 15 = 0
STEP
2
:
STEP
3
:
Pulling out like terms
3.1 Pull out like factors :
-4x2 - 16x - 15 = -1 • (4x2 + 16x + 15)
Trying to factor by splitting the middle term
3.2 Factoring 4x2 + 16x + 15
The first term is, 4x2 its coefficient is 4 .
The middle term is, +16x its coefficient is 16 .
The last term, "the constant", is +15
Step-1 : Multiply the coefficient of the first term by the constant 4 • 15 = 60
Step-2 : Find two factors of 60 whose sum equals the coefficient of the middle term, which is 16 .
-60 + -1 = -61
-30 + -2 = -32
-20 + -3 = -23
-15 + -4 = -19
-12 + -5 = -17
-10 + -6 = -16
-6 + -10 = -16
-5 + -12 = -17
-4 + -15 = -19
-3 + -20 = -23
-2 + -30 = -32
-1 + -60 = -61
1 + 60 = 61
2 + 30 = 32
3 + 20 = 23
4 + 15 = 19
5 + 12 = 17
6 + 10 = 16 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, 6 and 10
4x2 + 6x + 10x + 15
Step-4 : Add up the first 2 terms, pulling out like factors :
2x • (2x+3)
Add up the last 2 terms, pulling out common factors :
5 • (2x+3)
Step-5 : Add up the four terms of step 4 :
(2x+5) • (2x+3)
Which is the desired factorization
Equation at the end of step
3
:
(-2x - 3) • (2x + 5) = 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.
Solving a Single Variable Equation:
4.2 Solve : -2x-3 = 0
Add 3 to both sides of the equation :
-2x = 3
Multiply both sides of the equation by (-1) : 2x = -3
Divide both sides of the equation by 2:
x = -3/2 = -1.500
Solving a Single Variable Equation:
4.3 Solve : 2x+5 = 0
Subtract 5 from both sides of the equation :
2x = -5
Divide both sides of the equation by 2:
x = -5/2 = -2.500