If F ( 1 ) = 0 F(1)=0 F ( 1 ) = 0 , What Are All The Roots Of The Function F ( X ) = X 3 + 3 X 2 − X − 3 F(x)=x^3+3x^2-x-3 F ( X ) = X 3 + 3 X 2 − X − 3 ? Use The Remainder Theorem.A. X = − 1 , X = 1 X=-1, X=1 X = − 1 , X = 1 , Or X = 3 X=3 X = 3 B. X = − 3 , X = − 1 X=-3, X=-1 X = − 3 , X = − 1 , Or X = 1 X=1 X = 1 C. X = − 3 X=-3 X = − 3 Or X = 1 X=1 X = 1 D.

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Introduction

The Remainder Theorem is a fundamental concept in algebra that helps us find the roots of a polynomial function. It states that if we divide a polynomial $f(x)$ by a linear factor $(x-a)$, the remainder is equal to $f(a)$. In this article, we will use the Remainder Theorem to find the roots of the function $f(x)=x^3+3x^2-x-3$, given that $f(1)=0$.

Understanding the Remainder Theorem

The Remainder Theorem is a powerful tool that allows us to find the roots of a polynomial function by evaluating the function at specific values of $x$. The theorem states that if we divide a polynomial $f(x)$ by a linear factor $(x-a)$, the remainder is equal to $f(a)$. This means that if we know the remainder of the division, we can determine the value of $a$.

Applying the Remainder Theorem to the Given Function

We are given that $f(1)=0$, which means that the remainder of the division of $f(x)$ by $(x-1)$ is equal to $0$. Using the Remainder Theorem, we can write:

$$f(1) = 1^3 + 3(1)^2 - 1 - 3 = 0$$

This equation is true, which means that $(x-1)$ is a factor of $f(x)$. We can use this information to factorize the function $f(x)$.

Factoring the Function

Since $(x-1)$ is a factor of $f(x)$, we can write:

$$f(x) = (x-1)(x^2 + 4x + 3)$$

We can further factorize the quadratic expression $x^2 + 4x + 3$ as:

$$f(x) = (x-1)(x+1)(x+3)$$

Finding the Roots of the Function

Now that we have factored the function $f(x)$, we can find the roots of the function by setting each factor equal to zero. We have:

$$(x-1) = 0 \Rightarrow x = 1$$

$$(x+1) = 0 \Rightarrow x = -1$$

$$(x+3) = 0 \Rightarrow x = -3$$

Therefore, the roots of the function $f(x)$ are $x=-3$, $x=-1$, and $x=1$.

Conclusion

In this article, we used the Remainder Theorem to find the roots of the function $f(x)=x^3+3x^2-x-3$, given that $f(1)=0$. We applied the theorem to factorize the function and then found the roots of the function by setting each factor equal to zero. The roots of the function are $x=-3$, $x=-1$, and $x=1$.

Final Answer

The final answer is: $\boxed{B}$

$$$$

Introduction

The Remainder Theorem is a fundamental concept in algebra that helps us find the roots of a polynomial function. It states that if we divide a polynomial $f(x)$ by a linear factor $(x-a)$, the remainder is equal to $f(a)$. In this article, we will use the Remainder Theorem to find the roots of the function $f(x)=x^3+3x^2-x-3$, given that $f(1)=0$.

Understanding the Remainder Theorem

The Remainder Theorem is a powerful tool that allows us to find the roots of a polynomial function by evaluating the function at specific values of $x$. The theorem states that if we divide a polynomial $f(x)$ by a linear factor $(x-a)$, the remainder is equal to $f(a)$. This means that if we know the remainder of the division, we can determine the value of $a$.

Applying the Remainder Theorem to the Given Function

We are given that $f(1)=0$, which means that the remainder of the division of $f(x)$ by $(x-1)$ is equal to $0$. Using the Remainder Theorem, we can write:

$$f(1) = 1^3 + 3(1)^2 - 1 - 3 = 0$$

This equation is true, which means that $(x-1)$ is a factor of $f(x)$. We can use this information to factorize the function $f(x)$.

Factoring the Function

Since $(x-1)$ is a factor of $f(x)$, we can write:

$$f(x) = (x-1)(x^2 + 4x + 3)$$

We can further factorize the quadratic expression $x^2 + 4x + 3$ as:

$$f(x) = (x-1)(x+1)(x+3)$$

Finding the Roots of the Function

Now that we have factored the function $f(x)$, we can find the roots of the function by setting each factor equal to zero. We have:

$$(x-1) = 0 \Rightarrow x = 1$$

$$(x+1) = 0 \Rightarrow x = -1$$

$$(x+3) = 0 \Rightarrow x = -3$$

Therefore, the roots of the function $f(x)$ are $x=-3$, $x=-1$, and $x=1$.

Conclusion

In this article, we used the Remainder Theorem to find the roots of the function $f(x)=x^3+3x^2-x-3$, given that $f(1)=0$. We applied the theorem to factorize the function and then found the roots of the function by setting each factor equal to zero. The roots of the function are $x=-3$, $x=-1$, and $x=1$.

Q&A

Q: What is the Remainder Theorem?

A: The Remainder Theorem is a fundamental concept in algebra that helps us find the roots of a polynomial function. It states that if we divide a polynomial $f(x)$ by a linear factor $(x-a)$, the remainder is equal to $f(a)$.

Q: How do we apply the Remainder Theorem to find the roots of a function?

A: To apply the Remainder Theorem, we need to evaluate the function at specific values of $x$. If we know the remainder of the division, we can determine the value of $a$.

Q: What is the significance of the given condition $f(1)=0$?

A: The given condition $f(1)=0$ means that the remainder of the division of $f(x)$ by $(x-1)$ is equal to $0$. This information helps us to factorize the function $f(x)$.

Q: How do we factorize the function $f(x)$?

A: We can factorize the function $f(x)$ by using the information obtained from the Remainder Theorem. In this case, we have:

$$f(x) = (x-1)(x^2 + 4x + 3)$$

We can further factorize the quadratic expression $x^2 + 4x + 3$ as:

$$f(x) = (x-1)(x+1)(x+3)$$

Q: What are the roots of the function $f(x)$?

A: The roots of the function $f(x)$ are $x=-3$, $x=-1$, and $x=1$.

Q: How do we find the roots of the function $f(x)$?

A: We can find the roots of the function $f(x)$ by setting each factor equal to zero. We have:

$$(x-1) = 0 \Rightarrow x = 1$$

$$(x+1) = 0 \Rightarrow x = -1$$

$$(x+3) = 0 \Rightarrow x = -3$$

Therefore, the roots of the function $f(x)$ are $x=-3$, $x=-1$, and $x=1$.

Final Answer

The final answer is: $\boxed{B}$