How do you find all zeros of $f (x) = 5 x^{4} + 15 x^{2} + 10$ $f(x)=5x^4+15x^2+10$ ?

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How do you find all zeros of $f (x) = 5 x^{4} + 15 x^{2} + 10$ $f(x)=5x^4+15x^2+10$ ?

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Answer 1 · 2016-12-30T00:01:33

$f (x)$ $f(x)$ has zeros $\pm i$ $+-i$ and $\pm \sqrt{2} i$ $+-sqrt(2)i$

Explanation:

We will use the difference of squares identity, which can be written:

$a^{2} - b^{2} = (a - b) (a + b)$ $a^2-b^2 = (a-b)(a+b)$

with $a = x$ $a=x$ and $b = i$ $b=i$ or $b = \sqrt{2} i$ $b=sqrt(2)i$ as follows:

$f (x) = 5 x^{4} + 15 x^{2} + 10$ $f(x) = 5x^4+15x^2+10$

$f (x) = 5 (x^{4} + 3 x^{2} + 2)$ $color(white)(f(x)) = 5(x^4+3x^2+2)$

$f (x) = 5 (x^{2} + 1) (x^{2} + 2)$ $color(white)(f(x)) = 5(x^2+1)(x^2+2)$

$f (x) = 5 (x^{2} - i^{2}) (x^{2} - {(\sqrt{2} i)}^{2})$ $color(white)(f(x)) = 5(x^2-i^2)(x^2-(sqrt(2)i)^2)$

$f (x) = 5 (x - i) (x + i) (x - \sqrt{2} i) (x + \sqrt{2} i)$ $color(white)(f(x)) = 5(x-i)(x+i)(x-sqrt(2)i)(x+sqrt(2)i)$

Hence the zeros of $f (x)$ $f(x)$ are:

$x = \pm i$ $x = +-i$

$x = \pm \sqrt{2} i$ $x = +-sqrt(2)i$

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How do you find all zeros of $f (x) = 5 x^{4} + 15 x^{2} + 10$ $f(x)=5x^4+15x^2+10$ ?

1 Answer

Explanation:

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How do you find all zeros of f(x)=5x4+15x2+10f(x)=5x^4+15x^2+10?

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How do you find all zeros of $f (x) = 5 x^{4} + 15 x^{2} + 10$ $f(x)=5x^4+15x^2+10$ ?