Determine all the real zeros of the polynomial $P(x) = x^4 + 2x^3 - 2x^2 - 3x + 2$. Use the quadratic formula if necessary.
The leading coefficient of $P$ is $1$, so all the rational zeros are integers. They are the divisors of constant term $2$. Thus, the possible candidates are
$\pm 1, \pm 2$
Using Synthetic Division,
We find that $1$ is a zero and that $P$ factors as
$\displaystyle x^4 + 2x^3 - 2x^2 - 3x + 2 = (x - 1)\left(x^3 + 3x^2 + x - 2 \right)$
We now factor the quotient $x^3 + 3x^2 + x - 2$ and its possible zeros are
$ \pm 1, \pm 2$
Using Synthetic Division,
We find that $-1, 1$ and $2$ are not zeros but that $-2$ is a zero and that $P$ factors as
$x^4 + 2x^3 - 2x^2 - 3x + 2 = (x - 1)(x + 2)(x^2 + x - 1)$
We now factor the quotient $x^2 + x - 1$ using the quadratic formula
$
\begin{equation}
\begin{aligned}
x =& \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\\
\\
x =& \frac{-1 \pm \sqrt{(1)^2 - 4 (1)(-1)}}{2(1)}
\\
\\
x =& \frac{-1 \pm \sqrt{5}}{2}
\end{aligned}
\end{equation}
$
The zeros of $P$ are $\displaystyle 1, -2, \frac{-1 + \sqrt{5}}{2}$ and $\displaystyle \frac{-1 - \sqrt{5}}{2}$
Wednesday, August 27, 2014
College Algebra, Chapter 4, 4.4, Section 4.4, Problem 48
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