If $f(x)=x 4-x 3+x^2$ And $g(x)=-x^2$, Where $ X ≠ 0 X \neq 0 X  = 0 [/tex], What Is $(f / G)(x)$?A. $x^2-x+1$ B. $ X 2 + X + 1 X^2+x+1 X 2 + X + 1 [/tex] C. $-x^2+x-1$ D. $-x^2-x-1$

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Understanding the Problem

When dividing polynomials, it's essential to remember that the process is similar to long division in arithmetic. However, with polynomials, we need to consider the degree of the polynomials and the leading coefficients. In this problem, we are given two polynomials: $f(x)=x4-x3+x^2$ and $g(x)=-x^2$. Our goal is to find the quotient of these two polynomials, denoted as $(f/g)(x)$.

Recalling Polynomial Division Rules

Before we begin, let's recall some essential rules for polynomial division:

  • When dividing polynomials, we need to divide the leading term of the numerator by the leading term of the denominator.
  • The degree of the quotient is determined by the difference between the degrees of the numerator and the denominator.
  • When the degree of the numerator is less than the degree of the denominator, the quotient is a constant.

Dividing (f/g)(x)

Now, let's apply these rules to our problem. We need to divide $f(x)=x4-x3+x^2$ by $g(x)=-x^2$. To do this, we'll start by dividing the leading term of the numerator, $x^4$, by the leading term of the denominator, $-x^2$.

Step 1: Divide the Leading Terms

To divide $x^4$ by $-x^2$, we need to multiply $x^4$ by the reciprocal of $-x^2$, which is $-1/x^2$. This gives us $-x^2$.

Step 2: Multiply the Denominator by the Result

Next, we need to multiply the entire denominator, $-x^2$, by the result from Step 1, $-x^2$. This gives us $x^4$.

Step 3: Subtract the Result from the Numerator

Now, we need to subtract the result from Step 2, $x^4$, from the original numerator, $x4-x3+x^2$. This gives us $-x3+x2$.

Step 4: Repeat the Process

We need to repeat the process from Step 1, dividing the leading term of the new numerator, $-x^3$, by the leading term of the denominator, $-x^2$. This gives us $x$.

Step 5: Multiply the Denominator by the Result

Next, we need to multiply the entire denominator, $-x^2$, by the result from Step 4, $x$. This gives us $-x^3$.

Step 6: Subtract the Result from the Numerator

Now, we need to subtract the result from Step 5, $-x^3$, from the new numerator, $-x3+x2$. This gives us $x^2$.

Step 7: Repeat the Process

We need to repeat the process from Step 1, dividing the leading term of the new numerator, $x^2$, by the leading term of the denominator, $-x^2$. This gives us $-1$.

Step 8: Multiply the Denominator by the Result

Next, we need to multiply the entire denominator, $-x^2$, by the result from Step 7, $-1$. This gives us $x^2$.

Step 9: Subtract the Result from the Numerator

Now, we need to subtract the result from Step 8, $x^2$, from the new numerator, $x^2$. This gives us $0$.

The Quotient

Since the remainder is $0$, we can stop the division process. The quotient is the sum of the results from each step: $-x^2+x-1$.

Conclusion

In this article, we have shown the step-by-step process of dividing the polynomials $f(x)=x4-x3+x^2$ and $g(x)=-x^2$. By following the rules of polynomial division, we have found the quotient to be $-x^2+x-1$. This result is consistent with the options provided in the problem.

Final Answer

The final answer is: −x2+x−1\boxed{-x^2+x-1}

Q: What is polynomial division?

A: Polynomial division is a process of dividing one polynomial by another to obtain a quotient and a remainder. It is similar to long division in arithmetic, but with polynomials.

Q: What are the rules for polynomial division?

A: The rules for polynomial division are:

  • When dividing polynomials, we need to divide the leading term of the numerator by the leading term of the denominator.
  • The degree of the quotient is determined by the difference between the degrees of the numerator and the denominator.
  • When the degree of the numerator is less than the degree of the denominator, the quotient is a constant.

Q: How do I divide a polynomial by a binomial?

A: To divide a polynomial by a binomial, follow these steps:

  1. Divide the leading term of the numerator by the leading term of the denominator.
  2. Multiply the entire denominator by the result.
  3. Subtract the result from the numerator.
  4. Repeat the process until the remainder is zero.

Q: What is the difference between polynomial division and long division?

A: The main difference between polynomial division and long division is that polynomial division involves dividing polynomials, which can have multiple terms, whereas long division involves dividing numbers.

Q: Can I use a calculator to divide polynomials?

A: Yes, you can use a calculator to divide polynomials. However, it's essential to understand the process of polynomial division to ensure that you're using the calculator correctly.

Q: How do I check my work when dividing polynomials?

A: To check your work when dividing polynomials, follow these steps:

  1. Multiply the quotient by the denominator.
  2. Add the remainder to the product.
  3. The result should be equal to the numerator.

Q: What are some common mistakes to avoid when dividing polynomials?

A: Some common mistakes to avoid when dividing polynomials include:

  • Forgetting to divide the leading term of the numerator by the leading term of the denominator.
  • Not multiplying the entire denominator by the result.
  • Not subtracting the result from the numerator.
  • Not repeating the process until the remainder is zero.

Q: Can I divide a polynomial by a polynomial with a higher degree?

A: No, you cannot divide a polynomial by a polynomial with a higher degree. This is because the degree of the quotient would be negative, which is not possible.

Q: How do I simplify a quotient when dividing polynomials?

A: To simplify a quotient when dividing polynomials, follow these steps:

  1. Factor the numerator and denominator.
  2. Cancel out any common factors.
  3. Simplify the resulting expression.

Q: What is the remainder theorem?

A: The remainder theorem states that if a polynomial f(x) is divided by x - a, then the remainder is equal to f(a).

Q: How do I use the remainder theorem to find the remainder of a polynomial?

A: To use the remainder theorem to find the remainder of a polynomial, follow these steps:

  1. Substitute the value of a into the polynomial.
  2. Evaluate the expression.
  3. The result is the remainder.

Conclusion

In this article, we have answered some of the most frequently asked questions about dividing polynomials. We have covered topics such as the rules for polynomial division, how to divide a polynomial by a binomial, and how to check your work when dividing polynomials. We have also discussed common mistakes to avoid and how to simplify a quotient when dividing polynomials.