Math  /  Algebra

QuestionDrag the expressions below to express the rational expression in simplified terms. 3x+43x2+x4\frac{3 x+4}{3 x^{2}+x-4} 1 x1x-1 3 xx x+1-x+1 0

Studdy Solution

STEP 1

Assumptions
1. The given rational expression is 3x+43x2+x4\frac{3x + 4}{3x^2 + x - 4}.
2. We need to simplify this rational expression.
3. To simplify, we will factorize the numerator and the denominator, if possible.
4. We will then cancel out any common factors between the numerator and the denominator.

STEP 2

First, let's factorize the denominator 3x2+x43x^2 + x - 4. We need to find two numbers that multiply to 3(4)=123 \cdot (-4) = -12 and add up to 11.

STEP 3

The numbers that satisfy these conditions are 44 and 3-3, because 4(3)=124 \cdot (-3) = -12 and 4+(3)=14 + (-3) = 1.

STEP 4

Rewrite the middle term xx using 4x4x and 3x-3x:
3x2+x4=3x2+4x3x43x^2 + x - 4 = 3x^2 + 4x - 3x - 4

STEP 5

Group the terms to factor by grouping:
(3x2+4x)+(3x4)(3x^2 + 4x) + (-3x - 4)

STEP 6

Factor out the common factors from each group:
x(3x+4)1(3x+4)x(3x + 4) - 1(3x + 4)

STEP 7

Factor out the common binomial factor (3x+4)(3x + 4):
(3x+4)(x1)(3x + 4)(x - 1)

STEP 8

Now, the denominator 3x2+x43x^2 + x - 4 is factorized as (3x+4)(x1)(3x + 4)(x - 1).

STEP 9

Next, observe the numerator 3x+43x + 4. It is already in its simplest form.

STEP 10

Rewrite the original rational expression with the factorized form of the denominator:
3x+4(3x+4)(x1)\frac{3x + 4}{(3x + 4)(x - 1)}

STEP 11

Cancel out the common factor (3x+4)(3x + 4) in the numerator and the denominator:
3x+4(3x+4)(x1)=1x1\frac{3x + 4}{(3x + 4)(x - 1)} = \frac{1}{x - 1}

STEP 12

The simplified form of the given rational expression is:
1x1\frac{1}{x - 1}

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