Find the HCF: $x^3y+y^4, x^4+x^2y^2+y^4, 2ax^3-2ax^2y+2axy^2$.

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,

$1^{st} expression = x^3y+y^4$

= $y(x^3+y^3)$

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


$2^{nd} expression = x^4+x^2y^2+y^4$

= $x^4+y^4+x^2y^2$

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

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

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

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


$3^{rd} expression = 2ax^3-2ax^2y+2axy^2$

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

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

Explanation to the above answer.


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

Step 2: Take the common factor y and re-write the expression.

Step 3: Use the formula of (a³+b³) to find the factors of this expression. (a³+b³) = (a+b)(a²-ab+b²).

Step 4:  Write the second expression given in the question.

Step 5: Re-arrange the expression to make it in the form of a²+b²+a.

Step 6: (a²+b²) can be written as (a+b)²-2ab.

Step 7: -2x²y²+x²y² = -(xy)². And, we get the expression in the form of (a²-b²).

Step 8: Expand the expression (a²-b²) in factor form using (a²-b²) = (a+b)(a-b).

Step 9: Write the third expression given in the question.

Step 10: Take the common term 2ax as a common factor and re-write the expression.

Step 11: Write the H.C.F. (Highest Common Factor) of the given expressions by analysing the factors you generated in each expressions. Here, (x²-xy+y²) are the common factors.

Here is the Facebook link to the solution of this question in image. 

Here is the Website link to the guide of solving HCF and LCM.

Question: Find the HCF: x³y+y⁴, x⁴+x²y²+y⁴, 2ax³-2ax²y+2axy². | HCF and LCM | SciPiPupil

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