Use The Fact That $x = -2$ Is A Zero Of $f(x) = X^3 + 4x^2 - 187x - 382$ To Find The Other Zeros Of The Polynomial.Other Zeros:

Mar 13, 2025 by ADMIN 132 views

**Finding Other Zeros of a Polynomial Using a Given Zero**

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Introduction

In algebra, finding the zeros of a polynomial is a crucial step in understanding its behavior and properties. Given a polynomial function, we can use various methods to find its zeros, including factoring, synthetic division, and the Rational Root Theorem. In this article, we will explore how to find the other zeros of a polynomial using a given zero.

The Problem

We are given a polynomial function:

f(x) = x^3 + 4x^2 - 187x - 382

and we know that $x = -2$ is a zero of this polynomial. Our goal is to find the other zeros of the polynomial.

Using the Factor Theorem

The Factor Theorem states that if $f(a) = 0$, then $(x - a)$ is a factor of $f(x)$. In this case, we know that $f(-2) = 0$, so $(x + 2)$ is a factor of $f(x)$.

Dividing the Polynomial

To find the other zeros of the polynomial, we can divide $f(x)$ by $(x + 2)$ using polynomial long division or synthetic division. Let's use synthetic division to divide $f(x)$ by $(x + 2)$.

|  -2 | 1  4 -187 -382 |
|    | 1  6 -191  -2  |
|    | 0  2  -4  -2  |

The result of the division is:

f(x) = (x + 2)(x^2 + 2x - 191)

Finding the Other Zeros

Now that we have factored the polynomial, we can find the other zeros by setting the quadratic factor equal to zero and solving for $x$.

x^2 + 2x - 191 = 0

We can solve this quadratic equation using the quadratic formula:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

In this case, $a = 1$, $b = 2$, and $c = -191$. Plugging these values into the formula, we get:

x = \frac{-2 \pm \sqrt{2^2 - 4(1)(-191)}}{2(1)}

x = \frac{-2 \pm \sqrt{4 + 764}}{2}

x = \frac{-2 \pm \sqrt{768}}{2}

x = \frac{-2 \pm 28}{2}

So, the other zeros of the polynomial are:

x = \frac{-2 + 28}{2} = 13

x = \frac{-2 - 28}{2} = -15

Conclusion

In this article, we used the fact that $x = -2$ is a zero of the polynomial $f(x) = x^3 + 4x^2 - 187x - 382$ to find the other zeros of the polynomial. We used the Factor Theorem to divide the polynomial by $(x + 2)$ and then solved the resulting quadratic equation to find the other zeros. The other zeros of the polynomial are $x = 13$ and $x = -15$.

Other Zeros

Zero	Value
1	13
2	-15

References

Q: What is the Factor Theorem?

A: The Factor Theorem states that if $f(a) = 0$, then $(x - a)$ is a factor of $f(x)$. This means that if we know a zero of a polynomial, we can use it to find a factor of the polynomial.

Q: How do I use the Factor Theorem to find a factor of a polynomial?

A: To use the Factor Theorem, we need to know a zero of the polynomial. Once we have a zero, we can write the factor as $(x - a)$, where $a$ is the zero. For example, if we know that $x = -2$ is a zero of the polynomial $f(x) = x^3 + 4x^2 - 187x - 382$, we can write the factor as $(x + 2)$.

Q: What is synthetic division?

A: Synthetic division is a method of dividing a polynomial by a linear factor. It is a shortcut for polynomial long division and is often used to find the zeros of a polynomial.

Q: How do I use synthetic division to find the zeros of a polynomial?

A: To use synthetic division, we need to know a zero of the polynomial. Once we have a zero, we can use synthetic division to divide the polynomial by the linear factor. The result will be a quotient and a remainder. If the remainder is zero, then the linear factor is a factor of the polynomial.

Q: What is the quadratic formula?

A: The quadratic formula is a formula for solving quadratic equations of the form $ax^2 + bx + c = 0$. The formula is:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Q: How do I use the quadratic formula to find the zeros of a polynomial?

A: To use the quadratic formula, we need to know the coefficients of the quadratic equation. Once we have the coefficients, we can plug them into the formula and solve for $x$. The result will be two solutions, which are the zeros of the polynomial.

Q: What are the other zeros of the polynomial $f(x) = x^3 + 4x^2 - 187x - 382$?

A: We know that $x = -2$ is a zero of the polynomial. Using synthetic division, we can divide the polynomial by $(x + 2)$ and find the other zeros. The result is:

f(x) = (x + 2)(x^2 + 2x - 191)

Solving the quadratic equation $x^2 + 2x - 191 = 0$, we find that the other zeros are $x = 13$ and $x = -15$.

Q: How do I find the zeros of a polynomial using a graphing calculator?

A: To find the zeros of a polynomial using a graphing calculator, we can graph the polynomial and find the x-intercepts. The x-intercepts are the zeros of the polynomial.

Q: What are some common mistakes to avoid when finding the zeros of a polynomial?

A: Some common mistakes to avoid when finding the zeros of a polynomial include:

Not using the correct method for finding the zeros (e.g. using synthetic division instead of polynomial long division)
Not checking the remainder when using synthetic division
Not solving the quadratic equation correctly when using the quadratic formula
Not checking the solutions for extraneous solutions

Q: How do I check if a solution is an extraneous solution?

A: To check if a solution is an extraneous solution, we can plug the solution back into the original equation and check if it is true. If the solution is not true, then it is an extraneous solution.

Q: What are some real-world applications of finding the zeros of a polynomial?

A: Some real-world applications of finding the zeros of a polynomial include:

Modeling population growth and decline
Modeling the motion of objects
Finding the maximum or minimum value of a function
Solving optimization problems

Q: How do I find the maximum or minimum value of a function using the zeros of a polynomial?

A: To find the maximum or minimum value of a function using the zeros of a polynomial, we can use the fact that the maximum or minimum value occurs at a critical point. The critical points are the zeros of the derivative of the function.

Q: What is the derivative of a polynomial?

A: The derivative of a polynomial is a polynomial of one degree less than the original polynomial. For example, if we have a polynomial $f(x) = x^3 + 4x^2 - 187x - 382$, the derivative is $f'(x) = 3x^2 + 8x - 187$.

Q: How do I find the critical points of a function using the derivative?

A: To find the critical points of a function using the derivative, we can set the derivative equal to zero and solve for $x$. The result will be the critical points, which are the zeros of the derivative.

Q: What are some common mistakes to avoid when finding the critical points of a function?

A: Some common mistakes to avoid when finding the critical points of a function include:

Not using the correct method for finding the critical points (e.g. using synthetic division instead of polynomial long division)
Not checking the remainder when using synthetic division
Not solving the quadratic equation correctly when using the quadratic formula
Not checking the solutions for extraneous solutions

Q: How do I check if a solution is an extraneous solution when finding the critical points of a function?

A: To check if a solution is an extraneous solution when finding the critical points of a function, we can plug the solution back into the original equation and check if it is true. If the solution is not true, then it is an extraneous solution.

Q: What are some real-world applications of finding the critical points of a function?

A: Some real-world applications of finding the critical points of a function include:

Modeling population growth and decline
Modeling the motion of objects
Finding the maximum or minimum value of a function
Solving optimization problems

Q: How do I find the maximum or minimum value of a function using the critical points of a function?

A: To find the maximum or minimum value of a function using the critical points of a function, we can use the fact that the maximum or minimum value occurs at a critical point. The critical points are the zeros of the derivative of the function.

Q: What is the derivative of a function?

A: The derivative of a function is a function that represents the rate of change of the original function. For example, if we have a function $f(x) = x^3 + 4x^2 - 187x - 382$, the derivative is $f'(x) = 3x^2 + 8x - 187$.

Q: How do I find the derivative of a function?

A: To find the derivative of a function, we can use the power rule, the product rule, or the quotient rule. The power rule states that if we have a function $f(x) = x^n$, then the derivative is $f'(x) = nx^{n-1}$. The product rule states that if we have a function $f(x) = u(x)v(x)$, then the derivative is $f'(x) = u'(x)v(x) + u(x)v'(x)$. The quotient rule states that if we have a function $f(x) = \frac{u(x)}{v(x)}$, then the derivative is $f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{v(x)^2}$.

Q: What are some common mistakes to avoid when finding the derivative of a function?

A: Some common mistakes to avoid when finding the derivative of a function include:

Not using the correct method for finding the derivative (e.g. using synthetic division instead of polynomial long division)
Not checking the remainder when using synthetic division
Not solving the quadratic equation correctly when using the quadratic formula
Not checking the solutions for extraneous solutions

Q: How do I check if a solution is an extraneous solution when finding the derivative of a function?

A: To check if a solution is an extraneous solution when finding the derivative of a function, we can plug the solution back into the original equation and check if it is true. If the solution is not true, then it is an extraneous solution.

Q: What are some real-world applications of finding the derivative of a function?

A: Some real-world applications of finding the derivative of a function include:

Modeling population growth and decline
Modeling the motion of objects
Finding the maximum or minimum value of a function
Solving optimization problems