(2x^2+5x-3)/(x+3)

4 min read Jun 16, 2024

Simplifying the Rational Expression: (2x^2 + 5x - 3)/(x + 3)

This article will explore the simplification of the rational expression (2x^2 + 5x - 3)/(x + 3). We will use the method of polynomial long division to achieve this.

Understanding Rational Expressions

A rational expression is a fraction where the numerator and denominator are polynomials. Simplifying such expressions often involves finding common factors that can be canceled out.

Polynomial Long Division

Polynomial long division is a method used to divide polynomials, similar to long division with numbers. Here's how we apply it to our expression:

Set up the division:
```
    ________
x + 3 | 2x^2 + 5x - 3
```
Divide the leading terms:
- The leading term of the divisor (x + 3) is 'x'.
- The leading term of the dividend (2x^2 + 5x - 3) is '2x^2'.
- Dividing 2x^2 by x gives 2x.
- Write '2x' above the line.
```
    2x ______
x + 3 | 2x^2 + 5x - 3
```
Multiply the quotient by the divisor:
- Multiply '2x' by (x + 3), which gives 2x^2 + 6x.
- Write this result below the dividend.
```
    2x ______
x + 3 | 2x^2 + 5x - 3
        2x^2 + 6x 
```
Subtract:
- Subtract the result (2x^2 + 6x) from the dividend.
- Note: We are subtracting the entire expression, so we change the signs.
```
    2x ______
x + 3 | 2x^2 + 5x - 3
        2x^2 + 6x 
        -------
           -x - 3
```

Bring down the next term:

Bring down the '-3' from the dividend.

    2x ______
x + 3 | 2x^2 + 5x - 3
        2x^2 + 6x 
        -------
           -x - 3

Repeat steps 2-5:
- Divide the leading term of the new dividend (-x) by the leading term of the divisor (x), which gives -1.
- Multiply -1 by (x + 3), which gives -x - 3.
- Subtract this result from the current dividend.
```
    2x - 1 ______
x + 3 | 2x^2 + 5x - 3
        2x^2 + 6x 
        -------
           -x - 3 
           -x - 3 
           -----
              0
```
The Remainder:
- The remainder is 0, which means the division is complete.

Simplifying the Expression

The result of the division gives us the following:

(2x^2 + 5x - 3)/(x + 3) = 2x - 1

This is the simplified form of the original rational expression.

Conclusion

We have successfully simplified the rational expression (2x^2 + 5x - 3)/(x + 3) using polynomial long division. The result shows that the expression can be reduced to a simpler form, 2x - 1. This process can be applied to various rational expressions to make them more manageable for further operations.