Question

put it all together!
given the function $f(x) = x^3 - 13x^2 + 23x - 11$

1. use the rational root theorem to come up with a list of all the possible roots of $f(x)$.

1. use the factor theorem to determine one of the actual roots.

1. use synthetic division to divide $f(x)$ by the root you found in #2.

1. look at the quotient you found in #3. factor it to find the other two factors.

1. write the factored form: $f(x) = $
2. the zeros are

Explanation:

Step1: List possible rational roots

By Rational Root Theorem, possible roots are $\pm1, \pm11$.

Step2: Test for actual root

Test $x=1$: $f(1)=1^3 -13(1)^2 +23(1)-11=1-13+23-11=0$. So $x=1$ is a root.

Step3: Synthetic division with $x=1$

Set up synthetic division:

$$\begin{array}{r|rrrr} 1 & 1 & -13 & 23 & -11 \\ & & 1 & -12 & 11 \\ \hline & 1 & -12 & 11 & 0 \end{array}$$

Quotient: $x^2-12x+11$.

Step4: Factor the quadratic quotient

Factor $x^2-12x+11$: find two numbers that multiply to 11 and add to -12, which are -1 and -11. So $x^2-12x+11=(x-1)(x-11)$.

Step5: Write factored form

Combine the linear factors: $f(x)=(x-1)(x-1)(x-11)=(x-1)^2(x-11)$.

Step6: Identify all zeros

Set each factor equal to 0: $x-1=0$ gives $x=1$ (double root), $x-11=0$ gives $x=11$.

Answer:

1. Possible roots: $\pm1, \pm11$
2. Actual root: $x=1$
3. Quotient from synthetic division: $x^2-12x+11$
4. Factored quadratic: $(x-1)(x-11)$
5. Factored form: $f(x)=(x-1)^2(x-11)$
6. Zeros: $x=1$ (multiplicity 2), $x=11$