The Expression X 4 − 5 X 2 + 4 X + 14 X + 2 \frac{x^4-5x^2+4x+14}{x+2} X + 2 X 4 − 5 X 2 + 4 X + 14 ​ Is Equivalent To:1) $x 3-2x 2-x+6+\frac{2}{x+2}$2) $x^3-5x+4-\frac{14}{x+2}$3) $x 3+2x 2-x+2+\frac{18}{x+2}$4) X 3 + 2 X 2 − 9 X + 22 − 30 X + 2 X^3+2x^2-9x+22-\frac{30}{x+2} X 3 + 2 X 2 − 9 X + 22 − X + 2 30 ​

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Introduction

In algebra, simplifying complex expressions is a crucial skill that helps in solving equations and inequalities. One of the techniques used to simplify expressions is polynomial long division. In this article, we will explore the simplification of the expression $\frac{x^4-5x^2+4x+14}{x+2}$ using polynomial long division and identify its equivalent form from the given options.

Polynomial Long Division

Polynomial long division is a method used to divide a polynomial by another polynomial of lower degree. The process 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 the degree of the remainder is less than the degree of the divisor.

Simplifying the Expression

To simplify the expression $\frac{x^4-5x^2+4x+14}{x+2}$, we will use polynomial long division.

(x^4 - 5x^2 + 4x + 14) / (x + 2)

We start by dividing the highest degree term of the dividend, $x^4$, by the highest degree term of the divisor, $x$. This gives us $x^3$. We then multiply the entire divisor by $x^3$ and subtract it from the dividend.

x^3(x + 2) = x^5 + 2x^3
(x^4 - 5x^2 + 4x + 14) - (x^5 + 2x^3) = -5x^2 - 2x^3 + 4x + 14

Next, we divide the highest degree term of the new dividend, $-5x^2$, by the highest degree term of the divisor, $x$. This gives us $-5x$. We then multiply the entire divisor by $-5x$ and subtract it from the new dividend.

-5x(x + 2) = -5x^2 - 10x
(-5x^2 - 2x^3 + 4x + 14) - (-5x^2 - 10x) = -2x^3 + 14x + 14

We repeat the process by dividing the highest degree term of the new dividend, $-2x^3$, by the highest degree term of the divisor, $x$. This gives us $-2x^2$. We then multiply the entire divisor by $-2x^2$ and subtract it from the new dividend.

-2x^2(x + 2) = -2x^3 - 4x^2
(-2x^3 + 14x + 14) - (-2x^3 - 4x^2) = 14x + 4x^2 + 14

We repeat the process again by dividing the highest degree term of the new dividend, $14x$, by the highest degree term of the divisor, $x$. This gives us $14$. We then multiply the entire divisor by $14$ and subtract it from the new dividend.

14(x + 2) = 14x + 28
(14x + 4x^2 + 14) - (14x + 28) = 4x^2 - 14

Since the degree of the remainder, $4x^2 - 14$, is less than the degree of the divisor, $x + 2$, we stop the division process.

Identifying the Equivalent Form

The result of the polynomial long division is $x^3 - 2x^2 - 5x + 6$ with a remainder of $4x^2 - 14$. To express the original expression as a sum of the quotient and the remainder divided by the divisor, we write:

$$\frac{x^4-5x^2+4x+14}{x+2} = x^3 - 2x^2 - 5x + 6 + \frac{4x^2 - 14}{x+2}$$

However, we can simplify the remainder further by factoring out a common factor of $2$:

$$\frac{4x^2 - 14}{x+2} = \frac{2(2x^2 - 7)}{x+2}$$

Conclusion

In this article, we simplified the expression $\frac{x^4-5x^2+4x+14}{x+2}$ using polynomial long division and identified its equivalent form. The result of the polynomial long division is $x^3 - 2x^2 - 5x + 6$ with a remainder of $4x^2 - 14$. We then expressed the original expression as a sum of the quotient and the remainder divided by the divisor, and simplified the remainder further by factoring out a common factor of $2$.

Answer

The expression $\frac{x^4-5x^2+4x+14}{x+2}$ is equivalent to:

$$x^3 - 2x^2 - 5x + 6 + \frac{2}{x+2}$$

This is option 1) $x^3-2x^2-x+6+\frac{2}{x+2}$.

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Q: What is polynomial long division?

A: Polynomial long division is a method used to divide a polynomial by another polynomial of lower degree. The process 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 the degree of the remainder is less than the degree of the divisor.

Q: How do I simplify the expression $\frac{x^4-5x^2+4x+14}{x+2}$ using polynomial long division?

A: To simplify the expression $\frac{x^4-5x^2+4x+14}{x+2}$ using polynomial long division, you need to divide the highest degree term of the dividend, $x^4$, by the highest degree term of the divisor, $x$. This gives you $x^3$. You then multiply the entire divisor by $x^3$ and subtract it from the dividend. You repeat this process until the degree of the remainder is less than the degree of the divisor.

Q: What is the result of the polynomial long division?

A: The result of the polynomial long division is $x^3 - 2x^2 - 5x + 6$ with a remainder of $4x^2 - 14$.

Q: How do I express the original expression as a sum of the quotient and the remainder divided by the divisor?

A: To express the original expression as a sum of the quotient and the remainder divided by the divisor, you write:

$$\frac{x^4-5x^2+4x+14}{x+2} = x^3 - 2x^2 - 5x + 6 + \frac{4x^2 - 14}{x+2}$$

Q: Can I simplify the remainder further?

A: Yes, you can simplify the remainder further by factoring out a common factor of $2$:

$$\frac{4x^2 - 14}{x+2} = \frac{2(2x^2 - 7)}{x+2}$$

Q: What is the equivalent form of the expression $\frac{x^4-5x^2+4x+14}{x+2}$?

A: The equivalent form of the expression $\frac{x^4-5x^2+4x+14}{x+2}$ is:

$$x^3 - 2x^2 - 5x + 6 + \frac{2}{x+2}$$

Q: Which option is correct?

A: The correct option is 1) $x^3-2x^2-x+6+\frac{2}{x+2}$.

Q: What is the importance of simplifying expressions like $\frac{x^4-5x^2+4x+14}{x+2}$?

A: Simplifying expressions like $\frac{x^4-5x^2+4x+14}{x+2}$ is important because it helps in solving equations and inequalities. It also helps in understanding the properties of polynomials and their behavior.

Q: Can I use other methods to simplify the expression $\frac{x^4-5x^2+4x+14}{x+2}$?

A: Yes, you can use other methods to simplify the expression $\frac{x^4-5x^2+4x+14}{x+2}$, such as synthetic division or factoring. However, polynomial long division is a more straightforward method for this particular expression.

Q: How do I know when to use polynomial long division?

A: You should use polynomial long division when you need to divide a polynomial by another polynomial of lower degree. It is a useful technique for simplifying complex expressions and solving equations and inequalities.