If 2x + y = -5, prove that 8x^3 + y^3 - 30xy + 125 = 0
Answers
8/125x^3-27y^3
8/125x3-27y3
Final result :
(2x - 15y) • (4x2 + 30xy + 22
125
Step by step solution :
Step 1 :
Equation at the end of step 1 :
8
(——— • (x3)) - 33y3
125
Step 2 :
8
Simplify ———
125
Equation at the end of step 2 :
8
(——— • x3) - 33y3
125
Step 3 :
Equation at the end of step 3 :
8x3
——— - 33y3
125
Step 4 :
Rewriting the whole as an Equivalent Fraction :
4.1 Subtracting a whole from a fraction
Rewrite the whole as a fraction using 125 as the denominator :
33y3 33y3 • 125
33y3 = ———— = ——————————
1 125
Equivalent fraction : The fraction thus generated looks different but has the same value as the whole
Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator
Adding fractions that have a common denominator :
4.2 Adding up the two equivalent fractions
Add the two equivalent fractions which now have a common denominator
Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:
8x3 - (33y3 • 125) 8x3 - 3375y3
—————————————————— = ————————————
125 125
Trying to factor as a Difference of Cubes:
4.3 Factoring: 8x3 - 3375y3
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 : 8 is the cube of 2
Check : 3375 is the cube of 15
Check : x3 is the cube of x1
Check : y3 is the cube of y1
Factorization is :
(2x - 15y) • (4x2 + 30xy + 225y2)
Trying to factor a multi variable polynomial :
4.4 Factoring 4x2 + 30xy + 225y2
Try to factor this multi-variable trinomial using trial and error
Factorization fails
Final result :
(2x - 15y) • (4x2 + 30xy + 225y2)
Answer:
2x + y = -5
Cubing both the side,
8x^3+y^3+6xy(2x+y)= -125
8x^3+y^3-30xy+125=0
Hence proved.