(vii) 27x6 +64 (x + 1)3 factorise
Answers
Answer:
Step by Step Solution
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Rearrange:
Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :
27*x^6-(64*x^3)=0
Step by step solution :
STEP
1
:
Equation at the end of step 1
(27 • (x6)) - 26x3 = 0
STEP
2
:
Equation at the end of step
2
:
33x6 - 26x3 = 0
STEP
3
:
STEP
4
:
Pulling out like terms
4.1 Pull out like factors :
27x6 - 64x3 = x3 • (27x3 - 64)
Trying to factor as a Difference of Cubes:
4.2 Factoring: 27x3 - 64
Theory : A difference of two perfect cubes, a3 - b3 can be factored into
(a-b) • (a2 +ab +b2)
Proof : (a-b)•(a2+ab+b2) =
a3+a2b+ab2-ba2-b2a-b3 =
a3+(a2b-ba2)+(ab2-b2a)-b3 =
a3+0+0-b3 =
a3-b3
Check : 27 is the cube of 3
Check : 64 is the cube of 4
Check : x3 is the cube of x1
Factorization is :
(3x - 4) • (9x2 + 12x + 16)
Trying to factor by splitting the middle term
4.3 Factoring 9x2 + 12x + 16
The first term is, 9x2 its coefficient is 9 .
The middle term is, +12x its coefficient is 12 .
The last term, "the constant", is +16
Step-1 : Multiply the coefficient of the first term by the constant 9 • 16 = 144
Step-2 : Find two factors of 144 whose sum equals the coefficient of the middle term, which is 12 .
-144 + -1 = -145
-72 + -2 = -74
-48 + -3 = -51
-36 + -4 = -40
-24 + -6 = -30
-18 + -8 = -26
For tidiness, printing of 24 lines which failed to find two such factors, was suppressed
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Equation at the end of step
4
:
x3 • (3x - 4) • (9x2 + 12x + 16) = 0
STEP
5
:
Theory - Roots of a product
5.1 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:
5.2 Solve : x3 = 0
Solution is x3 = 0
Solving a Single Variable Equation:
5.3 Solve : 3x-4 = 0
Add 4 to both sides of the equation :
3x = 4
Divide both sides of the equation by 3:
x = 4/3 =