Divide: $ \left(20x^4 + 8x^3 - 19x^2 + 2x - 6\right) \div \left(-4x^2 + 3\right) }$Write Your Answer In The Following Form Quotient { + \frac{\text{Remainder }{-4x^2 + 3} $} . . . [ \frac{20x^4 + 8x^3 - 19x^2 + 2x - 6}{-4x^2 +

Introduction

When it comes to dividing polynomials, we often encounter complex expressions that require careful manipulation to simplify. In this article, we will explore the process of dividing a polynomial of degree 4 by a polynomial of degree 2, and we will use the long division method to achieve this. Our goal is to express the result in the form of a quotient plus a remainder, where the remainder is divided by the divisor.

The Long Division Method

The long division method is a step-by-step process that involves dividing the highest degree term of the dividend by the highest degree term of the divisor, and then multiplying the entire divisor by the result and subtracting it from the dividend. This process is repeated until we have a remainder that is of lower degree than the divisor.

Step 1: Divide the Highest Degree Term

To begin the long division process, we need to divide the highest degree term of the dividend, which is $20x^4$, by the highest degree term of the divisor, which is $-4x^2$. This gives us a quotient of $-5x^2$.

Step 2: Multiply the Divisor by the Quotient

Next, we multiply the entire divisor, $-4x^2 + 3$, by the quotient, $-5x^2$. This gives us a result of $20x^4 - 20x^2 + 15x^2 - 15$.

Step 3: Subtract the Result from the Dividend

We then subtract the result from the dividend, $20x^4 + 8x^3 - 19x^2 + 2x - 6$, to get a new expression: $8x^3 + x^2 + 2x - 21$.

Step 4: Repeat the Process

We repeat the process by dividing the highest degree term of the new expression, $8x^3$, by the highest degree term of the divisor, $-4x^2$. This gives us a quotient of $-2x$.

Step 5: Multiply the Divisor by the Quotient

We then multiply the entire divisor, $-4x^2 + 3$, by the quotient, $-2x$. This gives us a result of $-8x^3 + 6x$.

Step 6: Subtract the Result from the Dividend

We then subtract the result from the dividend, $8x^3 + x^2 + 2x - 21$, to get a new expression: $x^2 - 4x - 21$.

Step 7: Repeat the Process

We repeat the process by dividing the highest degree term of the new expression, $x^2$, by the highest degree term of the divisor, $-4x^2$. This gives us a quotient of $-\frac{1}{4}$.

Step 8: Multiply the Divisor by the Quotient

We then multiply the entire divisor, $-4x^2 + 3$, by the quotient, $-\frac{1}{4}$. This gives us a result of $-x^2 + \frac{3}{4}$.

Step 9: Subtract the Result from the Dividend

We then subtract the result from the dividend, $x^2 - 4x - 21$, to get a remainder of $-4x - \frac{93}{4}$.

The Result

The result of the long division process is a quotient of $-5x^2 - 2x - \frac{1}{4}$, plus a remainder of $-4x - \frac{93}{4}$, divided by the divisor $-4x^2 + 3$.

Conclusion

In this article, we have demonstrated the process of dividing a polynomial of degree 4 by a polynomial of degree 2 using the long division method. We have shown that the result can be expressed in the form of a quotient plus a remainder, where the remainder is divided by the divisor. This process is an essential tool in algebra and is used to simplify complex expressions and solve equations.

Final Answer

The final answer is: $\boxed{-5x^2 - 2x - \frac{1}{4} + \frac{-4x - \frac{93}{4}}{-4x^2 + 3}}$

Q&A: Dividing Polynomials

Q: What is the long division method for polynomials?

A: The long division method for polynomials is a step-by-step process that involves dividing the highest degree term of the dividend by the highest degree term of the divisor, and then multiplying the entire divisor by the result and subtracting it from the dividend. This process is repeated until we have a remainder that is of lower degree than the divisor.

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

A: To divide a polynomial by a binomial, we use the long division method. We divide the highest degree term of the dividend by the highest degree term of the divisor, and then multiply the entire divisor by the result and subtract it from the dividend. This process is repeated until we have a remainder that is of lower degree than the divisor.

Q: What is the remainder theorem?

A: The remainder theorem states that if we divide a polynomial f(x) by a binomial 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, we substitute the value of a into the polynomial f(x) and evaluate the result. This gives us the remainder of the polynomial.

Q: What is the difference between a quotient and a remainder?

A: A quotient is the result of dividing one polynomial by another, while a remainder is the amount left over after the division.

Q: How do I express the result of a polynomial division in the form of a quotient plus a remainder?

A: To express the result of a polynomial division in the form of a quotient plus a remainder, we write the quotient as the result of the division, and the remainder as the amount left over after the division.

Q: What is the significance of the remainder in polynomial division?

A: The remainder is significant because it tells us the amount left over after the division. It can also be used to determine the value of the polynomial at a specific point.

Q: How do I use polynomial division to solve equations?

A: To use polynomial division to solve equations, we divide the polynomial on one side of the equation by the polynomial on the other side. This gives us a new equation that we can solve.

Q: What are some common applications of polynomial division?

A: Polynomial division has many applications in mathematics and science, including solving equations, finding roots of polynomials, and simplifying complex expressions.

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

A: To check your work when dividing polynomials, you can use the following steps:

1. Multiply the divisor by the quotient and subtract the result from the dividend.
2. Check that the remainder is of lower degree than the divisor.
3. Check that the quotient is correct.

Conclusion

In this article, we have answered some common questions about dividing polynomials, including the long division method, the remainder theorem, and the significance of the remainder. We have also discussed some common applications of polynomial division and provided tips for checking your work.

Final Answer

The final answer is: $\boxed{-5x^2 - 2x - \frac{1}{4} + \frac{-4x - \frac{93}{4}}{-4x^2 + 3}}$