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Find the HCF of $16a^4 - 4a^2 + 4a -1$ and $8a^3+1$

My Attempt:

$2^{nd}$ expression $=8a^3+1$ $$(2a+1)(4a^2-2a+1)$$.

But, I could not factorize the first expresdion. Please help.

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    You don't need to factor the expressions. Just use [Eucid's algorithm](https://en.wikipedia.org/wiki/Polynomial_greatest_common_divisor). – 2017-02-08
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    @lhf, Could you please elaborate? – 2017-02-08
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    Please see [here](https://en.m.wikipedia.org/wiki/Polynomial_greatest_common_divisor) also. – 2017-02-08

2 Answers 2

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We have $$16a^4-4a^2 +4a -1 $$ $$ = 16a^4 -(4a^2 -4a +1) $$ $$=(4a^2)^2-(2a-1)^2$$ $$=(4a^2+2a-1)(4a^2-2a+1) $$

Hope it helps.

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You don't need to factor the expressions. Just use Eucid's algorithm.

$16a^4 - 4a^2 + 4a -1 = 2a(8a^3+1)- 4a^2 + 2a -1$

$8a^3+1 = (-2a-1)(- 4a^2 + 2a -1) + 0$

Therefore, the gcd is $- 4a^2 + 2a -1$.