Find the HCF: x⁴+4y⁴ and 2x³y+2xy³+4x²y² | HCF and LCM | SciPiPupil

Find the HCF: $x^4+4y^4$ and $2x^3y+4xy^3+4x^2y^2$.

This is a class 10 Question From H.C.F. chapter of Unit Algebra (Mathematics). All the steps for the solutions are mentioned in the description below. If that's hard for you to navigate, you can always visit the facebook link given at the end of every posts.

Solution:

Given,

$1^{st} expression = x^4+4y^4$

= $(x^2)^2 +(2y^2)^2$

= $(x^2+2y^2)^2 -2\cdot x^2 \cdot 2y^2$

= $(x^2+2y^2)^2 -4x^2y^2$

= $(x^2+2y^2)^2 -(2xy)^2$

= $(x^2+2y^2-2xy)(x^2+2y^2+2xy)$

$2^{nd} expression = (2x^3y+4xy^3+4x^2y^2)$

= $2xy(x^2+2y^2+2xy)$

Therefore, the HCF = $2xy(x^2+2y^2+2xy)$

Explanation to the above answer.

Step 1: Write the first expression given in the question.

Step 2: We have a⁴+b⁴ which we can write as (a²)²+(b²)².

Step 3:  We have a formula of (a²+b²) = (a+b)² -2ab.

Step 4: We perform the multiplication in the second term and write the obtained expression.

Step 5: The second term can be written with the exponent of two. So, we do this. Now, we obtain a²-b².

Step 6: We expand the formula of (a²-b²) = (a+b)(a-b).

Step 7: Write the second expression given in the question.

Step 8: Re-write the expression by taking 2xy as a common factor.

Step 9: Write the H.C.F. (Highest Common Factor) of the given expressions by analysing the factors you generated in each expressions. Here, (x²+2y²+2xy) are the common factors.

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Question: Find the HCF: x⁴+4y⁴ and 2x³y+2xy³+4x²y² | HCF and LCM | SciPiPupil

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