2+
D(x-3)(x+82)
a
b
IC
find a +7b
7b
5
7 b
Answers
Answer:
STEP
1
:
Equation at the end of step 1
((b6) + 7b3) - 8 = 0
STEP
2
:
Trying to factor by splitting the middle term
2.1 Factoring b6+7b3-8
The first term is, b6 its coefficient is 1 .
The middle term is, +7b3 its coefficient is 7 .
The last term, "the constant", is -8
Step-1 : Multiply the coefficient of the first term by the constant 1 • -8 = -8
Step-2 : Find two factors of -8 whose sum equals the coefficient of the middle term, which is 7 .
-8 + 1 = -7
-4 + 2 = -2
-2 + 4 = 2
-1 + 8 = 7 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -1 and 8
b6 - 1b3 + 8b3 - 8
Step-4 : Add up the first 2 terms, pulling out like factors :
b3 • (b3-1)
Add up the last 2 terms, pulling out common factors :
8 • (b3-1)
Step-5 : Add up the four terms of step 4 :
(b3+8) • (b3-1)
Which is the desired factorization
Trying to factor as a Sum of Cubes:
2.2 Factoring: b3+8
Theory : A sum 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 : 8 is the cube of 2
Check : b3 is the cube of b1
Factorization is :
(b + 2) • (b2 - 2b + 4)
Trying to factor by splitting the middle term
2.3 Factoring b2 - 2b + 4
The first term is, b2 its coefficient is 1 .
The middle term is, -2b its coefficient is -2 .
The last term, "the constant", is +4
Step-1 : Multiply the coefficient of the first term by the constant 1 • 4 = 4
Step-2 : Find two factors of 4 whose sum equals the coefficient of the middle term, which is -2 .
-4 + -1 = -5
-2 + -2 = -4
-1 + -4 = -5
1 + 4 = 5
2 + 2 = 4
4 + 1 = 5
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Trying to factor as a Difference of Cubes:
2.4 Factoring: b3-1
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 : 1 is the cube of 1
Check : b3 is the cube of b1
Factorization is :
(b - 1) • (b2 + b + 1)
Trying to factor by splitting the middle term
2.5 Factoring b2 + b + 1
The first term is, b2 its coefficient is 1 .
The middle term is, +b its coefficient is 1 .
The last term, "the constant", is +1
Step-1 : Multiply the coefficient of the first term by the constant 1 • 1 = 1
Step-2 : Find two factors of 1 whose sum equals the coefficient of the middle term, which is 1 .
-1 + -1 = -2
1 + 1 = 2
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Equation at the end of step
2
:
(b+2)•(b2-2b+4)•(b-1)•(b2+b+1) = 0
STEP
3
:
Theory - Roots of a product
3.1 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:
3.2 Solve : b+2 = 0
Subtract 2 from both sides of the equation :
b = -2
Step-by-step explanation:
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