Factorize: 54x^6y + 2x^3y^4
Answers
Final result :
2x3y2 • (27x3 - 19)
Step by step solution :
Step 1 :
Equation at the end of step 1 :
((54•(x6))•(y2))-((2•19x3)•y2)
Step 2 :
Equation at the end of step 2 :
((2•33x6) • y2) - (2•19x3y2)
Step 3 :
Step 4 :
Pulling out like terms :
4.1 Pull out like factors :
54x6y2 - 38x3y2 = 2x3y2 • (27x3 - 19)
Trying to factor as a Difference of Cubes:
4.2 Factoring: 27x3 - 19
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 : 19 is not a cube !!
Ruling : Binomial can not be factored as the difference of two perfect cubes
Polynomial Roots Calculator :
4.3 Find roots (zeroes) of : F(x) = 27x3 - 19
Polynomial Roots Calculator is a set of methods aimed at finding values of x for which F(x)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers x which can be expressed as the quotient of two integers
The Rational Root Theorem states that if a polynomial zeroes for a rational number P/Q then P is a factor of the Trailing Constant and Q is a factor of the Leading Coefficient
In this case, the Leading Coefficient is 27 and the Trailing Constant is -19.
The factor(s) are:
of the Leading Coefficient : 1,3 ,9 ,27
of the Trailing Constant : 1 ,19
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 -46.00
-1 3 -0.33 -20.00
-1 9 -0.11 -19.04
-1 27 -0.04 -19.00
-19 1 -19.00 -185212.00
-19 3 -6.33 -6878.00
-19 9 -2.11 -273.04
-19 27 -0.70 -28.41
1 1 1.00 8.00
1 3 0.33 -18.00
1 9 0.11 -18.96
1 27 0.04 -19.00
19 1 19.00 185174.00
19 3 6.33 6840.00
19 9 2.11 235.04
19 27 0.70 -9.59
Final result :
2x3y2 • (27x3 - 19)
Answer:
2x^3*y(3x+y)(9x^2 - 3xy +y^2)
Step-by-step explanation:
to factorize : 54x^6*y+2x^3*y^4
now take out the common factor that it 2x^3*y
=2x^3*y (27x^3+y^3)
=2x^3*y [(3x)^3 + y^3}
now the underlined part can be factorized using the (a^3 + b^3) identity
=2x^3*y (3x + y)(9x^2 - 3xy +y^2)
hope you get it..... :]