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Example 1
The complete factorization of $24x^4-6x^2$ is
- $x^2(24x^2-6)$
- $6x^2(4x^2-1)$
- $6x^2(2x-1)(2x+1)$
- $6(4x^4-x^2)$
- $x(24x^3-6x)$
Solution
- Complete factorization means no more factoring is possible. Start by factoring coefficients
- $24x^4-6x^2=6(4x^4-x^2)$
- Factor common terms with the variable to the same power
$=6x^2(4x^2-1)$
- We now look for any factoring inside the brackets. The term $4x^2-1$ could be written as ((2x)^2 - 1^2) which is the difference of two squares and can therefore be factored.(Note: $a^2-b^2=(a-b)(a+b)$). Hence
$=6x^2((2x)^2-1^2)= 6x^2(2x-1)(2x+1)$
- Note that there is no more factoring that can be done and therefore we have completely factored the given expression.
Answer C
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