If f(x) = x^3 + 4x^2 - 15x - 18 and x + 1 is a factor of f(x), then find all of the zeros of f(x) algebraically.
Solution
The problem asks to find all the zeros of f(x)=x3+4x2−15x−18 algebraically, given that x+1 is a factor of f(x).
Steps:
Factor Theorem: Since x+1 is a factor, we can apply synthetic division or long division of f(x) by x+1.
Synthetic Division:
Let's perform synthetic division using x=−1 (the root corresponding to x+1=0).
-1 & 1 & 4 & -15 & -18 \\
& & -1 & -3 & 18 \\
\hline
& 1 & 3 & -18 & 0 \\
\end{array}$$
The quotient is $$x^2 + 3x - 18$$ and the remainder is 0, which confirms that $$x + 1$$ is indeed a factor.
3. **Factor the Quadratic**: Now, solve the quadratic $$x^2 + 3x - 18 = 0$$ using the quadratic formula:
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
For $$a = 1$$, $$b = 3$$, and $$c = -18$$:
$$x = \frac{-3 \pm \sqrt{3^2 - 4(1)(-18)}}{2(1)} = \frac{-3 \pm \sqrt{9 + 72}}{2} = \frac{-3 \pm \sqrt{81}}{2}$$
$$x = \frac{-3 \pm 9}{2}$$
Thus, the two solutions are:
$$x = \frac{-3 + 9}{2} = 3 \quad \text{or} \quad x = \frac{-3 - 9}{2} = -6$$
4. **Final Zeros**: Therefore, the complete set of zeros of $$f(x)$$ is:
$$x = -1, \quad x = 3, \quad x = -6$$
Would you like more details on any step, or do you have any questions?
### Related questions:
1. How can we verify if a polynomial division was performed correctly?
2. How does the Factor Theorem help in finding polynomial roots?
3. What other methods can be used to factor cubic polynomials?
4. How does the quadratic formula apply to different types of equations?
5. Can all cubic polynomials be factored easily?
### Tip:
When factoring cubic polynomials, always check for simple factors (like $$x + 1$$) using the Factor Theorem before attempting more complex methods.