Math, asked by joyahamed4847, 1 year ago

Factorize: 54x^6y + 2x^3y^4

Answers

Answered by sonabrainly
6


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)



Answered by afreenshereef
16

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..... :]

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