Find all rational, irrational, and complex zeros (and state their multiplicities). $p ( x ) = x ^ { 3 } - x ^ { 2 } - 2 x + 2$
A) $\{ - \sqrt { 2 } i , \sqrt { 2 } i , 2 \}$
B) $\{ - \sqrt { 2 } i , \sqrt { 2 } i , 1 \}$
C) $\{ - 2,2,1 \}$
D) $\{ - \sqrt { 2 } , \sqrt { 2 } , 1 \}$
E) $( - 1 , - 2 \}$

Question

Quizplus · Accepted Answer

Since the degree of the polynomial is 3, we expect to find three zeros. By the Rational Root Theorem, any rational zero must be of the form $\pm\frac{p}{q}$, where $p$ is a factor of 2 and 2 is a factor of $q$. The possible rational zeros are then $\pm 1, \pm 2$, but testing these we find that none of them are roots. Therefore, there are no rational roots.

To find the remaining zeros, we can use the complex conjugate root theorem. If a polynomial $f(x)$ has real coefficients and $a+bi$ is a root (where $a,b$ are real and $i^2=-1$), then the conjugate $a-bi$ must also be a root. This means that any imaginary roots must come in conjugate pairs.

One way to factor this cubic polynomial is to use synthetic division to test the possible rational zeros, then factor the resulting quadratic polynomial using the quadratic formula. However, we can also try to spot a root by testing small values of $x$, or looking for symmetry in the equation.

If we graph the function, it appears to have a root around $x=1$, which turns out to be exactly at $x=1$ (by the Intermediate Value Theorem). We can use polynomial long division or synthetic division to divide $(x-1)$ into $x^3-x^2-2x+2$, or we can use the Factor Theorem to rewrite the polynomial as

$$x^3-x^2-2x+2 = (x-1)(x^2-2)\qquad (*)$$

and then use the quadratic formula to find the other two roots of the quadratic factor: $x=\pm\sqrt{2}$. These square roots are irrational, and since they appear in $(*)$ with multiplicity 1 (i.e. as simple roots), they each appear once in the list of zeros.

Therefore, the complete list of zeros (with multiplicities) is:

$$\{\sqrt{2},-\sqrt{2},1\}$$

which matches choice (D).

Find All Rational, Irrational, and Complex Zeros (And State Their p(x)=x3−x2−2x+2p ( x ) = x ^ { 3 } - x ^ { 2 } - 2 x + 2p(x)=x3−x2−2x+2

Find All Rational, Irrational, and Complex Zeros (And State Their $p ( x ) = x ^ { 3 } - x ^ { 2 } - 2 x + 2$