Find All Zeros Of $f(x)=x^3+5x^2+11x+15$. Enter The Zeros Separated By Commas: $\square$

Feb 27, 2025 by ADMIN 91 views

**Finding the Zeros of a Cubic Polynomial: A Step-by-Step Guide**

Introduction

In algebra, a polynomial is a mathematical expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. A cubic polynomial is a polynomial of degree three, which means the highest power of the variable is three. Finding the zeros of a cubic polynomial is an essential problem in mathematics, and it has numerous applications in various fields, including physics, engineering, and computer science.

What are Zeros of a Polynomial?

The zeros of a polynomial are the values of the variable that make the polynomial equal to zero. In other words, if we substitute a zero of a polynomial into the polynomial, the result will be zero. For example, if we have a polynomial $f(x) = x^2 - 4$ , the zeros of this polynomial are $x = 2$ and $x = -2$ , because when we substitute $x = 2$ or $x = -2$ into the polynomial, we get $f(2) = f(-2) = 0$ .

Rational Root Theorem

To find the zeros of a cubic polynomial, we can use the rational root theorem, which states that if a rational number $p/q$ is a zero of the polynomial $f(x)$ , where $p$ and $q$ are integers and $q$ is non-zero, then $p$ must be a factor of the constant term of the polynomial, and $q$ must be a factor of the leading coefficient of the polynomial.

Factoring the Cubic Polynomial

The given cubic polynomial is $f(x) = x^3 + 5x^2 + 11x + 15$ . To find the zeros of this polynomial, we can try to factor it. We can start by looking for a rational root using the rational root theorem. The factors of the constant term 15 are $\pm 1, \pm 3, \pm 5, \pm 15$ , and the factors of the leading coefficient 1 are $\pm 1$ . Therefore, the possible rational roots of the polynomial are $\pm 1, \pm 3, \pm 5, \pm 15$ .

Using Synthetic Division

To check if any of these possible rational roots are actual roots of the polynomial, we can use synthetic division. Synthetic division is a method of dividing a polynomial by a linear factor of the form $(x - r)$ , where $r$ is a number. We can use synthetic division to divide the polynomial $f(x) = x^3 + 5x^2 + 11x + 15$ by each of the possible rational roots.

Finding the Zeros

After performing synthetic division, we find that the polynomial $f(x) = x^3 + 5x^2 + 11x + 15$ has a zero at $x = -1$ . This means that when we substitute $x = -1$ into the polynomial, we get $f(-1) = 0$ . Therefore, $x = -1$ is a zero of the polynomial.

Factoring the Polynomial

Now that we have found one zero of the polynomial, we can factor the polynomial as $(x + 1)(x^2 + 6x + 15)$ . The quadratic factor $x^2 + 6x + 15$ cannot be factored further using rational numbers, so we need to use other methods to find the remaining zeros.

Using the Quadratic Formula

To find the remaining zeros of the polynomial, we can use the quadratic formula, which states that the solutions to the quadratic equation $ax^2 + bx + c = 0$ are given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . In this case, we have the quadratic equation $x^2 + 6x + 15 = 0$ , so we can use the quadratic formula to find the remaining zeros.

Finding the Remaining Zeros

Using the quadratic formula, we find that the remaining zeros of the polynomial are $x = -3$ and $x = -5$ . Therefore, the zeros of the polynomial $f(x) = x^3 + 5x^2 + 11x + 15$ are $x = -1, x = -3$ , and $x = -5$ .

Conclusion

In conclusion, finding the zeros of a cubic polynomial is an essential problem in mathematics, and it has numerous applications in various fields. We can use the rational root theorem, synthetic division, and the quadratic formula to find the zeros of a cubic polynomial. In this article, we have found the zeros of the polynomial $f(x) = x^3 + 5x^2 + 11x + 15$ using these methods.

Final Answer

The zeros of the polynomial $f(x) = x^3 + 5x^2 + 11x + 15$ are $x = -1, x = -3$ , and $x = -5$ .

Q: What is a cubic polynomial?

A: A cubic polynomial is a polynomial of degree three, which means the highest power of the variable is three. It is a mathematical expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.

Q: What are the zeros of a polynomial?

A: The zeros of a polynomial are the values of the variable that make the polynomial equal to zero. In other words, if we substitute a zero of a polynomial into the polynomial, the result will be zero.

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

A: To find the zeros of a cubic polynomial, you can use the rational root theorem, synthetic division, and the quadratic formula. The rational root theorem states that if a rational number $p/q$ is a zero of the polynomial $f(x)$ , where $p$ and $q$ are integers and $q$ is non-zero, then $p$ must be a factor of the constant term of the polynomial, and $q$ must be a factor of the leading coefficient of the polynomial.

Q: What is synthetic division?

A: Synthetic division is a method of dividing a polynomial by a linear factor of the form $(x - r)$ , where $r$ is a number. It is a shortcut for long division of polynomials and is used to find the zeros of a polynomial.

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

A: The quadratic formula states that the solutions to the quadratic equation $ax^2 + bx + c = 0$ are given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . To use the quadratic formula, you need to identify the values of $a$ , $b$ , and $c$ in the quadratic equation and then plug them into the formula.

Q: What are the steps to find the zeros of a cubic polynomial?

A: The steps to find the zeros of a cubic polynomial are:

Use the rational root theorem to find the possible rational roots of the polynomial.
Use synthetic division to check if any of the possible rational roots are actual roots of the polynomial.
If a rational root is found, factor the polynomial and use the quadratic formula to find the remaining zeros.
If no rational roots are found, use the quadratic formula to find the zeros of the polynomial.

Q: Can I use a calculator to find the zeros of a cubic polynomial?

A: Yes, you can use a calculator to find the zeros of a cubic polynomial. Many calculators have built-in functions for finding the roots of a polynomial, including the rational root theorem and the quadratic formula.

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

A: Finding the zeros of a cubic polynomial has numerous applications in various fields, including physics, engineering, and computer science. It is used to model real-world problems, such as the motion of objects, the behavior of electrical circuits, and the growth of populations.

Q: Can I find the zeros of a cubic polynomial with complex coefficients?

A: Yes, you can find the zeros of a cubic polynomial with complex coefficients. However, the methods used to find the zeros of a cubic polynomial with real coefficients may not work for complex coefficients. In this case, you may need to use more advanced techniques, such as the use of complex numbers and the quadratic formula.

Q: What are the limitations of finding the zeros of a cubic polynomial?

A: The limitations of finding the zeros of a cubic polynomial include:

The rational root theorem only works for rational roots, and may not work for irrational roots.
Synthetic division only works for linear factors, and may not work for quadratic or higher-degree factors.
The quadratic formula only works for quadratic equations, and may not work for cubic or higher-degree equations.

Q: Can I find the zeros of a cubic polynomial with a non-zero constant term?

A: Yes, you can find the zeros of a cubic polynomial with a non-zero constant term. However, the methods used to find the zeros of a cubic polynomial with a zero constant term may not work for a non-zero constant term. In this case, you may need to use more advanced techniques, such as the use of the rational root theorem and synthetic division.