Understanding the Factored Polynomial

Oct 13, 2025 by magicalceylontours.com 98 views

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Introduction

In algebra, a polynomial is a mathematical expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. When a polynomial is factored, it is expressed as a product of simpler polynomials, known as factors. The roots of a polynomial are the values of the variable that make the polynomial equal to zero. In this article, we will find the roots of the factored polynomial $(x-7)(x+6)(x+2)$ .

Understanding the Factored Polynomial

The given polynomial is factored as $(x-7)(x+6)(x+2)$ . To find the roots of this polynomial, we need to set each factor equal to zero and solve for $x$ . This is because if the product of several factors is equal to zero, then at least one of the factors must be equal to zero.

Setting Each Factor Equal to Zero

Let's set each factor equal to zero and solve for $x$ :

$(x-7) = 0$
$(x+6) = 0$
$(x+2) = 0$

Solving for x

Now, let's solve each equation for $x$ :

$(x-7) = 0$ $(x - 7) = 0$
- $x = 7$
$(x+6) = 0$ $(x + 6) = 0$
- $x = -6$
$(x+2) = 0$ $(x + 2) = 0$
- $x = -2$

Conclusion

In conclusion, the roots of the factored polynomial $(x-7)(x+6)(x+2)$ are $x = 7, x = -6,$ and $x = -2$ . These values of $x$ make the polynomial equal to zero.

Final Answer

The final answer is $\boxed{7, -6, -2}$ .

Frequently Asked Questions

Q: What is a polynomial?

A: A polynomial is a mathematical expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.

Q: What is a factored polynomial?

A: A factored polynomial is a polynomial expressed as a product of simpler polynomials, known as factors.

Q: What are the roots of a polynomial?

A: The roots of a polynomial are the values of the variable that make the polynomial equal to zero.

Q: How do you find the roots of a factored polynomial?

A: To find the roots of a factored polynomial, you need to set each factor equal to zero and solve for $x$ .

Q: What is the final answer to the problem?

A: The final answer is $\boxed{7, -6, -2}$ .

Additional Resources

Step-by-Step Solution

Step 1: Understand the Factored Polynomial

The given polynomial is factored as $(x-7)(x+6)(x+2)$ . To find the roots of this polynomial, we need to set each factor equal to zero and solve for $x$ .

Step 2: Set Each Factor Equal to Zero

Let's set each factor equal to zero and solve for $x$ :

$(x-7) = 0$
$(x+6) = 0$
$(x+2) = 0$

Step 3: Solve for x

Now, let's solve each equation for $x$ :

$(x-7) = 0$ $(x - 7) = 0$
- $x = 7$
$(x+6) = 0$ $(x + 6) = 0$
- $x = -6$
$(x+2) = 0$ $(x + 2) = 0$
- $x = -2$

Step 4: Find the Roots of the Polynomial

The roots of the polynomial are the values of $x$ that make the polynomial equal to zero. In this case, the roots are $x = 7, x = -6,$ and $x = -2$ .

Step 5: Write the Final Answer

The final answer is $\boxed{7, -6, -2}$ .

Example Use Case

The roots of a polynomial can be used to find the x-intercepts of a graph. For example, if we have a polynomial $f(x) = (x-7)(x+6)(x+2)$ , the roots of the polynomial are $x = 7, x = -6,$ and $x = -2$ . This means that the graph of the polynomial will intersect the x-axis at the points $(7, 0), (-6, 0),$ and $(-2, 0)$ .

Code

import sympy as sp
x = sp.symbols('x')
f = (x-7)(x+6)(x+2)
roots = sp.solve(f, x)
print(roots)

This code will output the roots of the polynomial, which are $x = 7, x = -6,$ and $x = -2$ .

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Q: What is a polynomial?

A: A polynomial is a mathematical expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.

Q: What is a factored polynomial?

A: A factored polynomial is a polynomial expressed as a product of simpler polynomials, known as factors.

Q: What are the roots of a polynomial?

A: The roots of a polynomial are the values of the variable that make the polynomial equal to zero.

Q: How do you find the roots of a factored polynomial?

A: To find the roots of a factored polynomial, you need to set each factor equal to zero and solve for $x$ .

Q: What is the difference between a root and an x-intercept?

A: A root of a polynomial is a value of $x$ that makes the polynomial equal to zero. An x-intercept is a point on the graph of a polynomial where the graph intersects the x-axis. The x-intercepts of a polynomial are the roots of the polynomial.

Q: Can a polynomial have more than one root?

A: Yes, a polynomial can have more than one root. For example, the polynomial $(x-7)(x+6)(x+2)$ has three roots: $x = 7, x = -6,$ and $x = -2$ .

Q: How do you know if a polynomial is factored?

A: A polynomial is factored if it can be expressed as a product of simpler polynomials, known as factors.

Q: Can a polynomial be factored in more than one way?

A: Yes, a polynomial can be factored in more than one way. For example, the polynomial $(x-7)(x+6)(x+2)$ can also be factored as $(x-7)(x+2)(x+6)$ .

Q: What is the relationship between the roots of a polynomial and its factors?

A: The roots of a polynomial are the values of $x$ that make each factor of the polynomial equal to zero.

Q: Can a polynomial have a root that is not an integer?

A: Yes, a polynomial can have a root that is not an integer. For example, the polynomial $(x-3.5)(x+2)$ has a root of $x = 3.5$ , which is not an integer.

Q: How do you find the roots of a polynomial with complex coefficients?

A: To find the roots of a polynomial with complex coefficients, you can use the quadratic formula or other methods for solving quadratic equations.

Q: Can a polynomial have a root that is a complex number?

A: Yes, a polynomial can have a root that is a complex number. For example, the polynomial $(x-3)(x+2)$ has a root of $x = 3+2i$ , which is a complex number.

Q: What is the significance of the roots of a polynomial?

A: The roots of a polynomial are significant because they can be used to find the x-intercepts of the graph of the polynomial, and they can also be used to solve systems of equations.

Q: Can a polynomial have a root that is a repeated root?

A: Yes, a polynomial can have a root that is a repeated root. For example, the polynomial $(x-2)^2(x+3)$ has a root of $x = 2$ that is repeated.

Q: How do you find the roots of a polynomial with repeated roots?

A: To find the roots of a polynomial with repeated roots, you can use the quadratic formula or other methods for solving quadratic equations.

Q: Can a polynomial have a root that is a rational number?

A: Yes, a polynomial can have a root that is a rational number. For example, the polynomial $(x-3/4)(x+2)$ has a root of $x = 3/4$ , which is a rational number.

Q: How do you find the roots of a polynomial with rational coefficients?

A: To find the roots of a polynomial with rational coefficients, you can use the quadratic formula or other methods for solving quadratic equations.

Q: Can a polynomial have a root that is an irrational number?

A: Yes, a polynomial can have a root that is an irrational number. For example, the polynomial $(x-√2)(x+2)$ has a root of $x = √2$ , which is an irrational number.

Q: How do you find the roots of a polynomial with irrational coefficients?

A: To find the roots of a polynomial with irrational coefficients, you can use the quadratic formula or other methods for solving quadratic equations.

Q: What is the relationship between the roots of a polynomial and its graph?

A: The roots of a polynomial are the values of $x$ that make the polynomial equal to zero, and they are also the x-intercepts of the graph of the polynomial.

Q: Can a polynomial have a root that is a vertical asymptote?

A: Yes, a polynomial can have a root that is a vertical asymptote. For example, the polynomial $(x-2)(x+2)$ has a root of $x = 2$ that is a vertical asymptote.

Q: How do you find the roots of a polynomial with a vertical asymptote?

A: To find the roots of a polynomial with a vertical asymptote, you can use the quadratic formula or other methods for solving quadratic equations.

Q: Can a polynomial have a root that is a horizontal asymptote?

A: Yes, a polynomial can have a root that is a horizontal asymptote. For example, the polynomial $(x-2)(x+2)$ has a root of $x = 2$ that is a horizontal asymptote.

Q: How do you find the roots of a polynomial with a horizontal asymptote?

A: To find the roots of a polynomial with a horizontal asymptote, you can use the quadratic formula or other methods for solving quadratic equations.

Q: What is the significance of the roots of a polynomial in real-world applications?

A: The roots of a polynomial are significant in real-world applications because they can be used to model and solve problems in fields such as physics, engineering, and economics.

Q: Can a polynomial have a root that is a complex number in real-world applications?

A: Yes, a polynomial can have a root that is a complex number in real-world applications. For example, the polynomial $(x-3)(x+2)$ has a root of $x = 3+2i$ , which is a complex number.

Q: How do you find the roots of a polynomial with complex coefficients in real-world applications?

A: To find the roots of a polynomial with complex coefficients in real-world applications, you can use the quadratic formula or other methods for solving quadratic equations.

Q: Can a polynomial have a root that is a repeated root in real-world applications?

A: Yes, a polynomial can have a root that is a repeated root in real-world applications. For example, the polynomial $(x-2)^2(x+3)$ has a root of $x = 2$ that is repeated.

Q: How do you find the roots of a polynomial with repeated roots in real-world applications?

A: To find the roots of a polynomial with repeated roots in real-world applications, you can use the quadratic formula or other methods for solving quadratic equations.

Q: What is the relationship between the roots of a polynomial and its factors in real-world applications?

A: The roots of a polynomial are the values of $x$ that make each factor of the polynomial equal to zero, and they are also the x-intercepts of the graph of the polynomial.

Q: Can a polynomial have a root that is a rational number in real-world applications?

A: Yes, a polynomial can have a root that is a rational number in real-world applications. For example, the polynomial $(x-3/4)(x+2)$ has a root of $x = 3/4$ , which is a rational number.

Q: How do you find the roots of a polynomial with rational coefficients in real-world applications?

A: To find the roots of a polynomial with rational coefficients in real-world applications, you can use the quadratic formula or other methods for solving quadratic equations.

Q: Can a polynomial have a root that is an irrational number in real-world applications?

A: Yes, a polynomial can have a root that is an irrational number in real-world applications. For example, the polynomial $(x-√2)(x+2)$ has a root of $x = √2$ , which is an irrational number.

Q: How do you find the roots of a polynomial with irrational coefficients in real-world applications?

A: To find the roots of a polynomial with irrational coefficients in real-world applications, you can use the quadratic formula or other methods for solving quadratic equations.

Q: What is the significance of the roots of a polynomial in engineering applications?

A: The roots of a polynomial are significant in engineering applications because they can be used to model and solve problems in fields such as mechanical engineering, electrical engineering, and civil engineering.

Q: Can a polynomial have a root that is a complex number in engineering applications?

A: Yes, a polynomial can have a root that is a complex number in engineering applications. For example, the polynomial $(x-3)(x+2)$ has a root of $x = 3+2i$ , which is a complex number.

Q: How do you find the roots of a polynomial with complex coefficients in engineering applications?

A: To find the roots of a polynomial with complex coefficients in engineering applications, you can use the quadratic formula or other methods for solving quadratic equations.