Finding Zeros Of A Polynomial: A Step-by-Step Guide

Hey guys! Today, we're diving into the exciting world of polynomials, specifically how to find their zeros. We'll tackle a problem where we're given a polynomial function, $f(x) = x^3 + 13x^2 + 44x + 32$, and told that $f(-1) = 0$. Our mission, should we choose to accept it, is to find all the zeros of $f(x)$ algebraically. Buckle up; let's get started!

Utilizing the Factor Theorem

Since we know that f(-1) = 0, the Factor Theorem comes to our rescue! This theorem tells us that if $f(c) = 0$ for some number 'c', then $(x - c)$ is a factor of $f(x)$. In our case, because $f(-1) = 0$, we know that $(x - (-1))$, which simplifies to $(x + 1)$, is a factor of our polynomial $f(x)$. This is a major breakthrough because it allows us to simplify the problem.

Now, how do we actually use this factor? We perform polynomial division! We'll divide $f(x) = x^3 + 13x^2 + 44x + 32$ by $(x + 1)$. Polynomial division might sound scary, but it's just a systematic way of breaking down the polynomial. Think of it like long division with numbers, but with variables. The goal is to find a quotient and a remainder. Ideally, the remainder should be zero, confirming that $(x + 1)$ is indeed a factor. If the remainder isn't zero, we've made a mistake somewhere, and we need to double-check our calculations. This step is important in finding the other roots of the polynomial function. Understanding polynomial division is really important in simplifying higher-degree polynomials.

After polynomial division, we obtain a quadratic expression. This is significantly easier to solve than the original cubic polynomial. The quadratic expression represents the remaining factor of the original polynomial. By finding the roots of this quadratic equation, we can determine the other zeros of the original polynomial. Methods such as factoring, completing the square, or using the quadratic formula are commonly employed to find these roots. Remember that complex roots can also exist, especially when the discriminant of the quadratic equation is negative. This step completes the process of finding all the zeros of the original cubic polynomial. By combining the root obtained from the Factor Theorem with the roots of the quadratic equation, we have a complete set of solutions.

Performing Polynomial Division

Let's perform the polynomial division of $x^3 + 13x^2 + 44x + 32$ by $(x + 1)$.

1. Divide the first term of the dividend ($x^3$) by the first term of the divisor ($x$). This gives us $x^2$.
2. Multiply the divisor $(x + 1)$ by $x^2$, resulting in $x^3 + x^2$.
3. Subtract this result from the dividend: $(x^3 + 13x^2 + 44x + 32) - (x^3 + x^2) = 12x^2 + 44x + 32$.
4. Bring down the next term, which is $+32$.
5. Divide the first term of the new expression ($12x^2$) by the first term of the divisor ($x$). This gives us $12x$.
6. Multiply the divisor $(x + 1)$ by $12x$, resulting in $12x^2 + 12x$.
7. Subtract this result from the current expression: $(12x^2 + 44x + 32) - (12x^2 + 12x) = 32x + 32$.
8. Divide the first term of the new expression ($32x$) by the first term of the divisor ($x$). This gives us $32$.
9. Multiply the divisor $(x + 1)$ by $32$, resulting in $32x + 32$.
10. Subtract this result from the current expression: $(32x + 32) - (32x + 32) = 0$.

So, the result of the division is $x^2 + 12x + 32$, with a remainder of $0$. This confirms that $(x + 1)$ is indeed a factor of $f(x)$. This meticulous process ensures accuracy and sets the stage for the next phase of our solution.

Solving the Quadratic Equation

Now that we've successfully divided the polynomial, we're left with the quadratic expression: $x^2 + 12x + 32$. To find the remaining zeros of $f(x)$, we need to solve this quadratic equation. There are a few ways we can do this:

• Factoring: We look for two numbers that multiply to 32 and add up to 12. Those numbers are 4 and 8. So, we can factor the quadratic as $(x + 4)(x + 8) = 0$.
• Quadratic Formula: If factoring isn't straightforward, we can use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. In our case, $a = 1$, $b = 12$, and $c = 32$. Plugging these values into the formula, we get the same solutions as factoring.
• Completing the Square: While this method works, it's generally more cumbersome for this particular quadratic.

Using the factoring method, we have $(x + 4)(x + 8) = 0$. This means either $(x + 4) = 0$ or $(x + 8) = 0$. Solving these equations, we find $x = -4$ and $x = -8$.

Therefore, the zeros of the quadratic equation are $x = -4$ and $x = -8$. These are also zeros of our original cubic polynomial, $f(x)$. Remember to always double-check your solutions by substituting them back into the original equation to ensure they satisfy the equation. This step ensures that you have correctly identified the zeros of the polynomial.

Listing All Zeros

We've found all the zeros! From the Factor Theorem, we knew that $x = -1$ is a zero. And by solving the quadratic equation, we found that $x = -4$ and $x = -8$ are also zeros. Therefore, the zeros of $f(x) = x^3 + 13x^2 + 44x + 32$ are $-1$, $-4$, and $-8$.

So, to recap, we started with a cubic polynomial and the knowledge of one of its zeros. We used the Factor Theorem to factor the polynomial, performed polynomial division to reduce it to a quadratic, solved the quadratic equation, and then listed all the zeros. Hopefully, this step-by-step guide has made the process clear and understandable. Keep practicing, and you'll become a pro at finding zeros of polynomials in no time!

In conclusion, finding the zeros of a polynomial algebraically involves using the Factor Theorem to identify a factor, performing polynomial division to reduce the polynomial to a lower degree, and then solving the resulting equation to find the remaining zeros. This methodical approach ensures that all zeros are identified accurately. Remember that complex roots can also be present, so it's important to consider all possibilities. Mastering these techniques will equip you with valuable skills in algebraic problem-solving.