Which Of The Following Functions Has A Factor Of $x+7$?A. $f(x)=x^3-x^2-65x-70$ B. $f(x)=x^3-15x^2+47x+63$ C. $f(x)=x^3-15x^2+47x+70$ D. $f(x)=x^3-x^2-65x-63$

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Introduction

In algebra, factoring polynomials is a crucial concept that helps us simplify complex expressions and solve equations. One of the most common methods of factoring is the factor theorem, which states that if a polynomial $f(x)$ has a factor of $(x-a)$, then $f(a) = 0$. In this article, we will explore how to determine which of the given functions has a factor of $x+7$.

Understanding the Factor Theorem

The factor theorem is a powerful tool in algebra that helps us determine the factors of a polynomial. If we know that a polynomial $f(x)$ has a factor of $(x-a)$, then we can conclude that $f(a) = 0$. This means that when we substitute $a$ into the polynomial, the result will be zero.

Applying the Factor Theorem to the Given Functions

To determine which of the given functions has a factor of $x+7$, we need to apply the factor theorem. We will substitute $-7$ into each of the functions and check if the result is zero.

Function A: $f(x)=x^3-x^2-65x-70$

import sympy as sp

x = sp.symbols('x')
f = x**3 - x**2 - 65*x - 70
result = f.subs(x, -7)
print(result)

When we substitute $-7$ into function A, we get:

$$f(-7) = (-7)^3 - (-7)^2 - 65(-7) - 70$$

Simplifying the expression, we get:

$$f(-7) = -343 - 49 + 455 - 70$$

$$f(-7) = -67$$

Since $f(-7) \neq 0$, we can conclude that function A does not have a factor of $x+7$.

Function B: $f(x)=x^3-15x^2+47x+63$

import sympy as sp

x = sp.symbols('x')
f = x**3 - 15*x**2 + 47*x + 63
result = f.subs(x, -7)
print(result)

When we substitute $-7$ into function B, we get:

$$f(-7) = (-7)^3 - 15(-7)^2 + 47(-7) + 63$$

Simplifying the expression, we get:

$$f(-7) = -343 - 735 - 329 + 63$$

$$f(-7) = -1344$$

Since $f(-7) \neq 0$, we can conclude that function B does not have a factor of $x+7$.

Function C: $f(x)=x^3-15x^2+47x+70$

import sympy as sp

x = sp.symbols('x')
f = x**3 - 15*x**2 + 47*x + 70
result = f.subs(x, -7)
print(result)

When we substitute $-7$ into function C, we get:

$$f(-7) = (-7)^3 - 15(-7)^2 + 47(-7) + 70$$

Simplifying the expression, we get:

$$f(-7) = -343 - 735 - 329 + 70$$

$$f(-7) = -1347$$

Since $f(-7) \neq 0$, we can conclude that function C does not have a factor of $x+7$.

Function D: $f(x)=x^3-x^2-65x-63$

import sympy as sp

x = sp.symbols('x')
f = x**3 - x**2 - 65*x - 63
result = f.subs(x, -7)
print(result)

When we substitute $-7$ into function D, we get:

$$f(-7) = (-7)^3 - (-7)^2 - 65(-7) - 63$$

Simplifying the expression, we get:

$$f(-7) = -343 - 49 + 455 - 63$$

$$f(-7) = 0$$

Since $f(-7) = 0$, we can conclude that function D has a factor of $x+7$.

Conclusion

In this article, we applied the factor theorem to determine which of the given functions has a factor of $x+7$. We substituted $-7$ into each of the functions and checked if the result was zero. We found that function D has a factor of $x+7$, while the other functions do not.

Final Answer

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Introduction

In our previous article, we explored how to determine which of the given functions has a factor of $x+7$ using the factor theorem. In this article, we will answer some frequently asked questions about factoring polynomials with the factor theorem.

Q: What is the factor theorem?

A: The factor theorem is a powerful tool in algebra that helps us determine the factors of a polynomial. If we know that a polynomial $f(x)$ has a factor of $(x-a)$, then we can conclude that $f(a) = 0$. This means that when we substitute $a$ into the polynomial, the result will be zero.

Q: How do I apply the factor theorem to a polynomial?

A: To apply the factor theorem to a polynomial, we need to substitute the value of $a$ into the polynomial and check if the result is zero. If the result is zero, then we can conclude that the polynomial has a factor of $(x-a)$.

Q: What is the significance of the factor theorem?

A: The factor theorem is significant because it helps us determine the factors of a polynomial. This is useful in solving equations and simplifying complex expressions.

Q: Can I use the factor theorem to factor a polynomial with multiple factors?

A: Yes, you can use the factor theorem to factor a polynomial with multiple factors. However, you need to apply the factor theorem multiple times, each time substituting a different value of $a$ into the polynomial.

Q: How do I determine the value of $a$ to substitute into the polynomial?

A: To determine the value of $a$ to substitute into the polynomial, you need to look for a value that makes the polynomial equal to zero. This value is called a root of the polynomial.

Q: Can I use the factor theorem to factor a polynomial with complex roots?

A: Yes, you can use the factor theorem to factor a polynomial with complex roots. However, you need to be careful when substituting complex values into the polynomial.

Q: What are some common mistakes to avoid when using the factor theorem?

A: Some common mistakes to avoid when using the factor theorem include:

• Not substituting the correct value of $a$ into the polynomial
• Not checking if the result is zero
• Not applying the factor theorem multiple times for polynomials with multiple factors

Q: How do I know if a polynomial has a factor of $x+7$?

A: To determine if a polynomial has a factor of $x+7$, you need to substitute $-7$ into the polynomial and check if the result is zero. If the result is zero, then you can conclude that the polynomial has a factor of $x+7$.

Conclusion

In this article, we answered some frequently asked questions about factoring polynomials with the factor theorem. We hope that this article has been helpful in understanding the concept of the factor theorem and how to apply it to factor polynomials.

Final Answer

The final answer is that the factor theorem is a powerful tool in algebra that helps us determine the factors of a polynomial. By substituting the correct value of $a$ into the polynomial and checking if the result is zero, we can conclude that the polynomial has a factor of $(x-a)$.