The Polynomial $f(x)$ Given Below Has -3 As A Zero:$f(x) = X^3 + 5x^2 - 16x + 30$Find All The Other Zeros Of $ F ( X ) F(x) F ( X ) [/tex].Provide Your Answer Below:

Mar 13, 2025 by ADMIN 170 views

**The Polynomial $f(x)$: Finding All Zeros**

Introduction

In mathematics, a polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. Polynomials are used to model various real-world phenomena, such as population growth, electrical circuits, and mechanical systems. In this article, we will focus on finding all the zeros of a given polynomial $f(x)$ , which has -3 as a known zero.

The Given Polynomial

The given polynomial is $f(x) = x^3 + 5x^2 - 16x + 30$ . We are told that -3 is a zero of this polynomial, which means that when we substitute $x = -3$ into the polynomial, the result is equal to zero.

Using Synthetic Division

To find the other zeros of the polynomial, we can use synthetic division. Synthetic division is a method for dividing a polynomial by a linear factor of the form $(x - c)$ , where $c$ is a constant. In this case, we know that $x + 3$ is a factor of the polynomial, since -3 is a zero.

Step 1: Write Down the Coefficients

We start by writing down the coefficients of the polynomial in descending order of powers of $x$ . The coefficients are 1, 5, -16, and 30.

Step 2: Perform Synthetic Division

Next, we perform synthetic division using the factor $x + 3$ . We write down the coefficients of the polynomial and the value of $c$ , which is -3.

	1	5	-16	30
-3	1	2	-14	0

We multiply the value of $c$ by the first coefficient and write the result below the line. Then, we add the second coefficient to the result and write the new value below the line. We repeat this process for the remaining coefficients.

Step 3: Find the Remaining Factor

After performing synthetic division, we obtain a new polynomial with coefficients 1, 2, -14, and 0. This polynomial can be written as $f(x) = (x + 3)(x^2 + 2x - 14)$ .

Factoring the Quadratic

To find the remaining zeros of the polynomial, we need to factor the quadratic expression $x^2 + 2x - 14$ . We can factor this expression by finding two numbers whose product is -14 and whose sum is 2.

Step 1: Find the Factors

The factors of -14 are 1, -1, 2, -2, 7, -7, 14, and -14. We need to find two numbers whose product is -14 and whose sum is 2.

Step 2: Factor the Quadratic

After trying different combinations of factors, we find that the quadratic expression can be factored as $(x + 7)(x - 2)$ .

Finding the Remaining Zeros

Now that we have factored the quadratic expression, we can find the remaining zeros of the polynomial. The zeros of the polynomial are the values of $x$ that make the polynomial equal to zero.

Step 1: Set Each Factor Equal to Zero

We set each factor equal to zero and solve for $x$ . For the first factor, we have $x + 7 = 0$ , which gives $x = -7$ . For the second factor, we have $x - 2 = 0$ , which gives $x = 2$ .

Step 2: List the Zeros

The zeros of the polynomial are $x = -3, x = -7$ , and $x = 2$ .

Conclusion

In this article, we used synthetic division and factoring to find all the zeros of a given polynomial $f(x)$ , which has -3 as a known zero. We found that the zeros of the polynomial are $x = -3, x = -7$ , and $x = 2$ . This demonstrates the importance of using algebraic techniques to solve polynomial equations and find the zeros of a polynomial.

Applications of Zeros

The zeros of a polynomial have many applications in mathematics and science. For example, the zeros of a polynomial can be used to model population growth, electrical circuits, and mechanical systems. In addition, the zeros of a polynomial can be used to find the maximum or minimum value of a function.

Real-World Examples

Here are some real-world examples of how the zeros of a polynomial are used:

Population Growth: The zeros of a polynomial can be used to model population growth. For example, the population of a city can be modeled using a polynomial equation, and the zeros of the polynomial can be used to find the maximum population.
Electrical Circuits: The zeros of a polynomial can be used to model electrical circuits. For example, the voltage across a resistor can be modeled using a polynomial equation, and the zeros of the polynomial can be used to find the maximum voltage.
Mechanical Systems: The zeros of a polynomial can be used to model mechanical systems. For example, the motion of a pendulum can be modeled using a polynomial equation, and the zeros of the polynomial can be used to find the maximum amplitude.

Conclusion

In conclusion, the zeros of a polynomial have many applications in mathematics and science. The zeros of a polynomial can be used to model population growth, electrical circuits, and mechanical systems. In addition, the zeros of a polynomial can be used to find the maximum or minimum value of a function. By using algebraic techniques to solve polynomial equations and find the zeros of a polynomial, we can gain a deeper understanding of the world around us.

Introduction

In our previous article, we explored the concept of zeros of a polynomial and how to find them using synthetic division and factoring. In this article, we will answer some frequently asked questions about the zeros of a polynomial.

Q: What is a zero of a polynomial?

A: A zero of a polynomial is a value of $x$ that makes the polynomial equal to zero. In other words, if we substitute $x = a$ into the polynomial, the result is equal to zero.

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

A: There are several methods to find the zeros of a polynomial, including synthetic division, factoring, and the quadratic formula. The method you choose will depend on the complexity of the polynomial and your personal preference.

Q: What is synthetic division?

A: Synthetic division is a method for dividing a polynomial by a linear factor of the form $(x - c)$ , where $c$ is a constant. It is a quick and efficient way to find the zeros of a polynomial.

Q: How do I perform synthetic division?

A: To perform synthetic division, you need to write down the coefficients of the polynomial in descending order of powers of $x$ . Then, you multiply the value of $c$ by the first coefficient and write the result below the line. You add the second coefficient to the result and write the new value below the line. You repeat this process for the remaining coefficients.

Q: What is factoring?

A: Factoring is a method for finding the zeros of a polynomial by expressing it as a product of simpler polynomials. It is a powerful tool for solving polynomial equations.

Q: How do I factor a polynomial?

A: To factor a polynomial, you need to find two numbers whose product is the constant term and whose sum is the coefficient of the linear term. You can then write the polynomial as a product of two binomials.

Q: What is the quadratic formula?

A: The quadratic formula is a method for finding the zeros of a quadratic polynomial of the form $ax^2 + bx + c$ . It is a powerful tool for solving quadratic equations.

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to plug in the values of $a$ , $b$ , and $c$ into the formula and simplify. The result will be the zeros of the polynomial.

Q: What are the applications of zeros of a polynomial?

A: The zeros of a polynomial have many applications in mathematics and science. They can be used to model population growth, electrical circuits, and mechanical systems. They can also be used to find the maximum or minimum value of a function.

Q: How do I use the zeros of a polynomial to model population growth?

A: To use the zeros of a polynomial to model population growth, you need to express the population as a polynomial function of time. Then, you can use the zeros of the polynomial to find the maximum population.

Q: How do I use the zeros of a polynomial to model electrical circuits?

A: To use the zeros of a polynomial to model electrical circuits, you need to express the voltage across a resistor as a polynomial function of time. Then, you can use the zeros of the polynomial to find the maximum voltage.

Q: How do I use the zeros of a polynomial to model mechanical systems?

A: To use the zeros of a polynomial to model mechanical systems, you need to express the motion of a pendulum as a polynomial function of time. Then, you can use the zeros of the polynomial to find the maximum amplitude.

Conclusion

In conclusion, the zeros of a polynomial are an important concept in mathematics and science. They can be used to model population growth, electrical circuits, and mechanical systems. By using algebraic techniques to solve polynomial equations and find the zeros of a polynomial, we can gain a deeper understanding of the world around us.

Frequently Asked Questions

Here are some frequently asked questions about the zeros of a polynomial:

Q: What is a zero of a polynomial? A: A zero of a polynomial is a value of $x$ that makes the polynomial equal to zero.
Q: How do I find the zeros of a polynomial? A: There are several methods to find the zeros of a polynomial, including synthetic division, factoring, and the quadratic formula.
Q: What is synthetic division? A: Synthetic division is a method for dividing a polynomial by a linear factor of the form $(x - c)$ , where $c$ is a constant.
Q: How do I perform synthetic division? A: To perform synthetic division, you need to write down the coefficients of the polynomial in descending order of powers of $x$ . Then, you multiply the value of $c$ by the first coefficient and write the result below the line. You add the second coefficient to the result and write the new value below the line. You repeat this process for the remaining coefficients.
Q: What is factoring? A: Factoring is a method for finding the zeros of a polynomial by expressing it as a product of simpler polynomials.
Q: How do I factor a polynomial? A: To factor a polynomial, you need to find two numbers whose product is the constant term and whose sum is the coefficient of the linear term. You can then write the polynomial as a product of two binomials.
Q: What is the quadratic formula? A: The quadratic formula is a method for finding the zeros of a quadratic polynomial of the form $ax^2 + bx + c$ .
Q: How do I use the quadratic formula? A: To use the quadratic formula, you need to plug in the values of $a$ , $b$ , and $c$ into the formula and simplify. The result will be the zeros of the polynomial.

Glossary

Here are some key terms related to the zeros of a polynomial:

Zero: A value of $x$ that makes the polynomial equal to zero.
Synthetic division: A method for dividing a polynomial by a linear factor of the form $(x - c)$ , where $c$ is a constant.
Factoring: A method for finding the zeros of a polynomial by expressing it as a product of simpler polynomials.
Quadratic formula: A method for finding the zeros of a quadratic polynomial of the form $ax^2 + bx + c$ .
Polynomial: An expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.