Question 964458
here's how I would do it:
{{{ f(x) = x^4 + 23x^2 + 22 }}}
Let {{{ z = x^2 }}}
Substituting:
{{{ f(x) = z^2 + 23z + 22 }}}
I can see that the factors for
finding roots
{{{ ( z + 22 )*( z + 1 ) = 0 }}}
The solutions for {{{ Z }}} are:
{{{ z = -22 }}}
{{{ z = -1 }}}
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So, now I can say:
{{{ x^2 = -22 }}}
{{{ x^2 = -1 }}}
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{{{ x = sqrt(22)*i }}}
{{{ x = -sqrt(22)*i }}}
{{{ x = i }}}
{{{ x = -i }}}
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The 4 factors of {{{ f(x) }}} are:
{{{ f(x) = ( x + sqrt(22)*i )*( x - sqrt(22)*i )*( x + i )*( x - i ) }}}
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check:
{{{ ( x + sqrt(22)*i )*( x - sqrt(22)*i ) }}}
{{{ x^2 + 22 }}}
and
{{{ ( x + i )*( x - i ) = x^2 + 1 ) }}}
and
{{{ f(x) = ( x^2 + 22 )*( x^2 + 1 ) }}}
{{{ f(x) = x^4 + 22x^2 + x^2 + 22 }}}
{{{ f(x) = x^4 + 23x^2 + 22 }}}
OK