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Find the HCF: $16x^4-4x^2-4x-1$ and $8x^3-1$.

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,

$1^{st} expression = 16x^4-4x^2-4x-1$

= $16x^4-(4x^2 +4x+1)$

= $16x^4 -\{(2x)^2 +2\cdot 2x\cdot 1 +1^2)\}$

= $(4x^2)^2 -(2x+1)^2$

= $\{4x^2+(2x+1)\}\{4x^2-(2x+1)\}$

= $(4x^2+2x+1)(4x^2-2x-1)$



$2^(nd) expression = 8x^3-1$

= $(2x)^3 -1^3$

= $(2x-1)(4x^2+2x+1)$


Therefore, the HCF = $(4x^2+2x+1)$

16x⁴-4x²-4x-1 and 8x³-1



Explanation to the above answer.


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

Step 2: We need to take the minus sign common from the last three terms as this will help us to further factorize the formula. 

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

Step 4: So, we write the formula in square form as (a+b)².

Step 5: Since we have a²-b² in step 4. We further write this as (a+b)(a-b). This is the factor formula of a²-b².

Step 6: As we had two terms in place of in (a+b)(a-b), we multiply the term 'b' with '+' and '-' sign respectively. 

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

Step 8: Write the terms in the form of cube.

Step 9: Write the factor formula of a³-b³) in the form of (a-b)(a²+ab+b²).

Step 10: Write the H.C.F. (Highest Common Factor) of the given expressions by analysing the factors you generated in each expressions. Here, (4x²+2x+1) are the common factors.

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Question: Find the HCF: 16x⁴-4x²-4x-1 and 8x³-1. | HCF and LCM | SciPiPupil

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