How do you factor $a^5 - 3a^4 + a^3$ ?

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Answer 1 · 2015-06-22T17:10:59

$a^5-3a^4+a^3$

$=a^3(a^2-3a+1)$

$=a^3(a-(3+sqrt(5))/2)(a-(3-sqrt(5))/2)$

First separate out the common factor $a^3$ to get:

$a^5-3a^4+a^3=a^3(a^2-3a+1)$

To find the factors of $f(a)=a^2-3a+1$ , use the quadratic formula to find the roots of $f(a) = 0$ , as:

$a = (3+-sqrt(3^2-(4xx1xx1)))/(2xx1)$

$=(3+-sqrt(9-4))/2$

$=(3+-sqrt(5))/2$

Hence $f(a)$ can be factored as:

$f(a) = (a-(3+sqrt(5))/2)(a-(3-sqrt(5))/2)$

How do you factor a^5 - 3a^4 + a^3a^5 - 3a^4 + a^3?