Find The Roots Of The Factored Polynomial $(x-4)(x+8)$.Write Your Answer As A List Of Values Separated By Commas.$x = \square, \square$

Feb 26, 2025 by ADMIN 138 views

Finding the Roots of a Factored Polynomial

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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 explore how to find the roots of a factored polynomial, using the example of the polynomial $(x-4)(x+8)$ .

What are Roots of a Polynomial?

The roots of a polynomial are the values of the variable that make the polynomial equal to zero. In other words, if we substitute a root of a polynomial into the polynomial, the result will be zero. For example, if we have a polynomial $p(x)$ and a root $r$ , then $p(r) = 0$ . The roots of a polynomial are also known as the solutions or the zeros of the polynomial.

Factoring a Polynomial

Factoring a polynomial involves expressing it as a product of simpler polynomials, known as factors. The factors of a polynomial are polynomials that, when multiplied together, give the original polynomial. For example, the polynomial $(x-4)(x+8)$ can be factored into two linear factors: $(x-4)$ and $(x+8)$ . These factors are the building blocks of the polynomial, and they can be used to find the roots of the polynomial.

Finding the Roots of a Factored Polynomial

To find the roots of a factored polynomial, we need to set each factor equal to zero and solve for the variable. In the case of the polynomial $(x-4)(x+8)$ , we have two factors: $(x-4)$ and $(x+8)$ . We can set each factor equal to zero and solve for $x$ :

$(x-4) = 0 \Rightarrow x = 4$
$(x+8) = 0 \Rightarrow x = -8$

Therefore, the roots of the polynomial $(x-4)(x+8)$ are $x = 4$ and $x = -8$ .

Conclusion

In conclusion, finding the roots of a factored polynomial involves setting each factor equal to zero and solving for the variable. By using this method, we can find the roots of a polynomial, which are the values of the variable that make the polynomial equal to zero. In this article, we used the example of the polynomial $(x-4)(x+8)$ to illustrate how to find the roots of a factored polynomial.

Example Use Cases

Finding the roots of a polynomial has many practical applications in mathematics and science. Here are a few examples:

Solving equations: Finding the roots of a polynomial can help us solve equations that involve polynomials. For example, if we have an equation $p(x) = 0$ , where $p(x)$ is a polynomial, we can find the roots of the polynomial to solve the equation.
Graphing functions: Finding the roots of a polynomial can help us graph functions that involve polynomials. For example, if we have a function $f(x)$ that involves a polynomial, we can find the roots of the polynomial to graph the function.
Optimization problems: Finding the roots of a polynomial can help us solve optimization problems that involve polynomials. For example, if we have an optimization problem that involves a polynomial, we can find the roots of the polynomial to solve the problem.

Final Thoughts

In conclusion, finding the roots of a factored polynomial is an important concept in algebra that has many practical applications. By using the method of setting each factor equal to zero and solving for the variable, we can find the roots of a polynomial, which are the values of the variable that make the polynomial equal to zero. We hope that this article has provided a clear and concise explanation of how to find the roots of a factored polynomial.

References

[1] "Algebra" by Michael Artin
[2] "Calculus" by Michael Spivak
[3] "Linear Algebra" by Jim Hefferon

FAQs

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 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 I find the roots of a factored polynomial? A: To find the roots of a factored polynomial, set each factor equal to zero and solve for the variable.

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Q: What is a polynomial?

A polynomial is a mathematical expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. Polynomials can be classified into different types, such as linear, quadratic, cubic, and so on, based on the degree of the polynomial.

Q: What are the roots of a polynomial?

The roots of a polynomial are the values of the variable that make the polynomial equal to zero. In other words, if we substitute a root of a polynomial into the polynomial, the result will be zero. The roots of a polynomial are also known as the solutions or the zeros of the polynomial.

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

To find the roots of a factored polynomial, set each factor equal to zero and solve for the variable. For example, if we have a factored polynomial $(x-4)(x+8)$ , we can set each factor equal to zero and solve for $x$ :

$(x-4) = 0 \Rightarrow x = 4$
$(x+8) = 0 \Rightarrow x = -8$

Therefore, the roots of the polynomial $(x-4)(x+8)$ are $x = 4$ and $x = -8$ .

Q: What is the difference between a root and a solution?

A root and a solution are often used interchangeably, but technically, a root refers to the value of the variable that makes the polynomial equal to zero, while a solution refers to the value of the variable that satisfies the equation.

Q: Can a polynomial have multiple roots?

Yes, a polynomial can have multiple roots. For example, the polynomial $(x-4)^2(x+8)$ has two roots: $x = 4$ and $x = 4$ . This is because the factor $(x-4)^2$ has a repeated root at $x = 4$ .

Q: How do I find the roots of a polynomial with multiple factors?

To find the roots of a polynomial with multiple factors, set each factor equal to zero and solve for the variable. For example, if we have a polynomial $(x-4)^2(x+8)$ , we can set each factor equal to zero and solve for $x$ :

$(x-4)^2 = 0 \Rightarrow x = 4$
$(x+8) = 0 \Rightarrow x = -8$

Therefore, the roots of the polynomial $(x-4)^2(x+8)$ are $x = 4$ and $x = -8$ .

Q: Can a polynomial have complex roots?

Yes, a polynomial can have complex roots. Complex roots are roots that involve imaginary numbers, such as $i = \sqrt{-1}$ . For example, the polynomial $(x^2+1)$ has two complex roots: $x = i$ and $x = -i$ .

Q: How do I find the roots of a polynomial with complex factors?

To find the roots of a polynomial with complex factors, set each factor equal to zero and solve for the variable. For example, if we have a polynomial $(x^2+1)$ , we can set each factor equal to zero and solve for $x$ :

$x^2+1 = 0 \Rightarrow x = i$ and $x = -i$

Therefore, the roots of the polynomial $(x^2+1)$ are $x = i$ and $x = -i$ .

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

Finding the roots of a polynomial is an important concept in algebra that has many practical applications. By finding the roots of a polynomial, we can solve equations, graph functions, and optimize problems that involve polynomials.

Q: Can you provide examples of how to find the roots of a polynomial?

Yes, here are some examples of how to find the roots of a polynomial:

Example 1: Find the roots of the polynomial $(x-4)(x+8)$ $(x - 4) (x + 8)$ .
- $(x-4) = 0 \Rightarrow x = 4$
- $(x+8) = 0 \Rightarrow x = -8$ Therefore, the roots of the polynomial $(x-4)(x+8)$ are $x = 4$ and $x = -8$ .
Example 2: Find the roots of the polynomial $(x^2+1)$ $(x^{2} + 1)$ .
- $x^2+1 = 0 \Rightarrow x = i$ and $x = -i$ Therefore, the roots of the polynomial $(x^2+1)$ are $x = i$ and $x = -i$ .

Q: Can you provide more examples of how to find the roots of a polynomial?

Yes, here are some more examples of how to find the roots of a polynomial:

Example 3: Find the roots of the polynomial $(x-3)^2(x+2)$ $(x - 3)^{2} (x + 2)$ .
- $(x-3)^2 = 0 \Rightarrow x = 3$
- $(x+2) = 0 \Rightarrow x = -2$ Therefore, the roots of the polynomial $(x-3)^2(x+2)$ are $x = 3$ and $x = -2$ .
Example 4: Find the roots of the polynomial $(x^2-4)$ $(x^{2} - 4)$ .
- $x^2-4 = 0 \Rightarrow x = \pm 2$ Therefore, the roots of the polynomial $(x^2-4)$ are $x = 2$ and $x = -2$ .

Q: Can you provide a summary of how to find the roots of a polynomial?

Yes, here is a summary of how to find the roots of a polynomial:

Set each factor equal to zero and solve for the variable.
Use the quadratic formula to find the roots of a quadratic polynomial.
Use the factoring method to find the roots of a polynomial with multiple factors.
Use the complex conjugate root theorem to find the roots of a polynomial with complex factors.

By following these steps, you can find the roots of a polynomial and solve equations, graph functions, and optimize problems that involve polynomials.