Find the HCF: $x^3-1, x^4+x^2+1, x^3+1+2x^2+2x$.

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,

$1^{st} expression = x^3-1$

= $(x-1)(x^2+x+1)$


$2^{nd} expression = x^4+x^2+1$

= $x^4+1+x^2$

= $(x^2)^2+1^2 +x^2$

= $(x^2+1)^2-2x^2+x^2$

= $(x^2+1)^2 -(x)^2$

= $(x^2+x+1)(x^2-x+1)$


$3^{rd} expression = x^3+1+2x^2+2x$

= $(x+1)(x^2-x+1) +2x(x+1)$

= $(x+1)\{(x^2-x+1+2x)\}$

= $(x+1)(x^2+x+1)$


Therefore, the HCF = $(x^2+x+1)$

Explanation to the above answer.


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

Step 2: Use the formula of (a³-b³) to find the factors of this expression. (a³-b³) = (a-b)(a²+ab+b²).

Step 3:  Write the second expression given in the question.

Step 4: Re-arrange the expression to make it in the form of a²+b²+a.

Step 5: (a²+b²) can be written as (a+b)²-2ab.

Step 6: -2x²+x² = -x². And, we get the expression in the form of (a²-b²).

Step 7: Expand the expression (a²-b²) in factor form using (a²-b²) = (a+b)(a-b).

Step 8: Write the third expression given in the question.

Step 9: Write the terms (a³+b³) in factor form using (a+b)(a²-ab+b²). And, write the remaining terms by taking the common factor.

Step 10: As we get (x+1) as common factor in the terms before and after the addition sign, we can re-write the expression by taking the term common.

Step 11: Write the answer after performing simple maths..

Step 12: Write the H.C.F. (Highest Common Factor) of the given expressions by analysing the factors you generated in each expressions. Here, (x²+x+1) are the common factors.

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Question: Find the HCF: x³-1, x⁴+x²+1, x³+1+2x²+2x. | HCF and LCM | SciPiPupil

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