How can you factor $f (x) = x^{4} - 12 x^{3} + 59 x^{2} - 138 x + 130$ $f(x)=x^4-12x^3+59x^2-138x+130$ ?

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How can you factor $f (x) = x^{4} - 12 x^{3} + 59 x^{2} - 138 x + 130$ $f(x)=x^4-12x^3+59x^2-138x+130$ ?

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Answer 1 · 2017-10-17T22:30:17

$x^{4} - 12 x^{3} + 59 x^{2} - 138 x + 130$ $x^4-12x^3+59x^2-138x+130$

$= (x^{2} - 6 x + 10) (x^{2} - 6 x + 13)$ $=(x^2-6x+10)(x^2-6x+13)$

$= (x - 3 - i) (x - 3 + i) (x - 3 - 2 i) (x - 3 + 2 i)$ $=(x-3-i)(x-3+i)(x-3-2i)(x-3+2i)$

Explanation:

Given:

$f (x) = x^{4} - 12 x^{3} + 59 x^{2} - 138 x + 130$ $f(x) = x^4-12x^3+59x^2-138x+130$

Simplify the quartic by noting the similarity to ${(x - 3)}^{4}$ $(x-3)^4$ ...

$x^{4} - 12 x^{3} + 59 x^{2} - 138 x + 130$ $x^4-12x^3+59x^2-138x+130$

$= x^{4} - 12 x^{3} + 54 x^{2} - 108 x + 81 + 5 x^{2} - 30 x + 45 + 4$ $=x^4-12x^3+54x^2-108x+81+5x^2-30x+45+4$

$= {(x - 3)}^{4} + 5 {(x - 3)}^{2} + 4$ $=(x-3)^4+5(x-3)^2+4$

$= ({(x - 3)}^{2} + 1) ({(x - 3)}^{2} + 4) = (x^{2} - 6 x + 10) (x^{2} - 6 x + 13)$ $=((x-3)^2+1)((x-3)^2+4) color(grey)(= (x^2-6x+10)(x^2-6x+13))$

$= ({(x - 3)}^{2} - i^{2}) ({(x - 3)}^{2} - {(2 i)}^{2})$ $=((x-3)^2-i^2)((x-3)^2-(2i)^2)$

$= (x - 3 - i) (x - 3 + i) (x - 3 - 2 i) (x - 3 + 2 i)$ $=(x-3-i)(x-3+i)(x-3-2i)(x-3+2i)$

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How can you factor $f (x) = x^{4} - 12 x^{3} + 59 x^{2} - 138 x + 130$ $f(x)=x^4-12x^3+59x^2-138x+130$ ?

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Explanation:

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How can you factor f(x)=x4−12x3+59x2−138x+130f(x)=x^4-12x^3+59x^2-138x+130?

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Explanation:

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How can you factor $f (x) = x^{4} - 12 x^{3} + 59 x^{2} - 138 x + 130$ $f(x)=x^4-12x^3+59x^2-138x+130$ ?