Multiply the rational expression and write the product in simplest form: x^2+2x−3/x^2+3x−4 ⋅ x^2+6x+8/x^2−9 .(1 point)
1. -2/3
2. (x+4)(x+2)/(x+3)(x-3)
3. x+2/x-3
4. -(x+2)/x+3
To multiply the rational expressions, we first need to factor each expression:
The numerator of the first rational expression, x^2 + 2x -3, factors to (x + 3)(x - 1).
The denominator of the first rational expression, x^2 + 3x - 4, factors to (x + 4)(x - 1).
The numerator of the second rational expression, x^2 + 6x +8, factors to (x + 2)(x + 4).
The denominator of the second rational expression, x^2 - 9, factors to (x - 3)(x + 3).
Next, we can cancel out some common factors:
(x + 3)(x - 1)/(x + 4)(x - 1) * (x + 2)(x + 4)/(x - 3)(x + 3)
The (x - 1) term in the numerator and denominator cancels out.
(x + 3)/(x + 4) * (x + 2)(x + 4)/(x - 3)
Now, we can multiply the numerators and denominators:
(x + 3)(x + 2)(x + 4)/(x + 4)(x - 3)
The (x + 4) term in the numerator and denominator cancels out.
(x + 3)(x + 2)/(x - 3)
So, the product of the rational expressions, in simplest form, is (x + 3)(x + 2)/(x - 3). Therefore, the correct answer is option 3.
To multiply the rational expressions and write the product in simplest form, we can follow these steps:
Step 1: Factor both the numerators and the denominators.
For the first rational expression, we have:
x^2 + 2x - 3 = (x + 3)(x - 1)
For the second rational expression, we have:
x^2 + 6x + 8 = (x + 4)(x + 2)
x^2 - 9 = (x + 3)(x - 3)
Step 2: Cancel out any common factors between the numerators and denominators.
Now, we can write the original rational expressions with the factored forms:
(x + 3)(x - 1)/(x + 3)(x - 3) * (x + 4)(x + 2)/(x + 3)(x - 3)
Step 3: Multiply the numerators and the denominators.
Multiplying the numerators: (x + 3)(x - 1)(x + 4)(x + 2)
Multiplying the denominators: (x + 3)(x - 3)(x + 4)(x + 2)
Step 4: Simplify the expression.
Cancel out any common factors between the numerator and the denominator:
(x + 3) cancels out, and (x + 4)(x + 2) cancels out.
The simplified expression is:
(x - 1)/(x - 3)
Therefore, the correct answer is 3. x - 1/x - 3.
To multiply rational expressions and write the product in simplest form, you need to follow these steps:
Step 1: Factor both the numerators and denominators completely.
The numerator of the first rational expression, x^2 + 2x - 3, can be factored as (x + 3)(x - 1), and the denominator, x^2 + 3x - 4, can be factored as (x + 4)(x - 1).
The numerator of the second rational expression, x^2 + 6x + 8, cannot be factored further, and the denominator, x^2 - 9, can be factored as (x + 3)(x - 3).
Step 2: Cancel out common factors between the numerator and the denominator.
Now, let's multiply both expressions together and write the product in simplest form:
((x + 3)(x - 1) / (x + 4)(x - 1)) * (x^2 + 6x + 8) / (x + 3)(x - 3)
Next, cancel out common factors from the numerators and denominators:
(x + 3) / (x + 4) * (x + 3)(x - 1) / (x - 3)
Step 3: Simplify the expression.
Expanding the numerator, we get:
(x^2 + 2x - 3)(x - 1) = x^3 - x^2 - x + 3x^2 + 2x - 3 = x^3 + 2x^2 + x - 3
The expression becomes:
(x^3 + 2x^2 + x - 3) / (x + 4)(x - 3)
So, the answer in simplest form is (x^3 + 2x^2 + x - 3) / (x + 4)(x - 3).