QuestionQuestion 6
Simplify: $\frac{4 x^{2}+9 x-9}{4 x^{2}+13 x+3} \div \frac{4 x^{2}+17 x-15}{x^{2}+3 x-10}$ Question Help: Video
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Studdy Solution

STEP 1

1. We are given a division of two rational expressions.
2. Simplifying involves converting the division into multiplication by the reciprocal.
3. Factoring each polynomial will help simplify the expression.

STEP 2

1. Rewrite the division as multiplication by the reciprocal.
2. Factor each polynomial in the expression.
3. Simplify the expression by canceling common factors.

STEP 3

Rewrite the division as multiplication by the reciprocal:
$\frac{4x^2 + 9x - 9}{4x^2 + 13x + 3} \div \frac{4x^2 + 17x - 15}{x^2 + 3x - 10} = \frac{4x^2 + 9x - 9}{4x^2 + 13x + 3} \times \frac{x^2 + 3x - 10}{4x^2 + 17x - 15}$

STEP 4

Factor each polynomial:
- Factor $4x^2 + 9x - 9$ . - Factor $4x^2 + 13x + 3$ . - Factor $x^2 + 3x - 10$ . - Factor $4x^2 + 17x - 15$ .
Let's begin with each factorization:
- $4x^2 + 9x - 9$ factors to $(4x - 3)(x + 3)$ . - $4x^2 + 13x + 3$ factors to $(4x + 1)(x + 3)$ . - $x^2 + 3x - 10$ factors to $(x + 5)(x - 2)$ . - $4x^2 + 17x - 15$ factors to $(4x - 3)(x + 5)$ .

STEP 5

Substitute the factored forms into the expression:
$\frac{(4x - 3)(x + 3)}{(4x + 1)(x + 3)} \times \frac{(x + 5)(x - 2)}{(4x - 3)(x + 5)}$
Cancel the common factors:
- $(4x - 3)$ cancels with $(4x - 3)$ . - $(x + 3)$ cancels with $(x + 3)$ . - $(x + 5)$ cancels with $(x + 5)$ .
The simplified expression is:
$\frac{x - 2}{4x + 1}$

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