Multiply the rational expression and write the product in the simplest form: (x^(2)+2x-3)/(x^(2)+3x-4)+(x^(2)+6x+8)/(x^(2)-9)
To multiply the rational expression, we must first factorize the denominators:
Denominator 1: x^2 + 3x - 4 = (x + 4)(x - 1)
Denominator 2: x^2 - 9 = (x + 3)(x - 3)
Therefore, the expression becomes:
[(x^2 + 2x - 3)/(x + 4)(x - 1)] * [(x^2 + 6x + 8)/(x + 3)(x - 3)]
Next, we multiply the numerators and denominators:
[(x^2 + 2x - 3)(x^2 + 6x + 8)] / [(x + 4)(x - 1)(x + 3)(x - 3)]
Expanding the numerators:
[(x^4 + 8x^3 + 22x^2 + 16x - 24)] / [(x + 4)(x - 1)(x + 3)(x - 3)]
Simplifying the expression further is not possible in this case.
To multiply the rational expression, we first need to find a common denominator for both expressions. The denominators are (x^2+3x-4) and (x^2-9).
To find the common denominator, we need to factor both denominators:
(x^2+3x-4) = (x+4)(x-1)
(x^2-9) = (x+3)(x-3)
The common denominator is then (x+4)(x-1)(x+3)(x-3).
Now, let's rewrite the original expression with the common denominator:
[(x^2+2x-3)/(x^2+3x-4)] + [(x^2+6x+8)/(x^2-9)]
= [(x^2+2x-3)(x+3)(x-3) + (x^2+6x+8)(x+4)(x-1)] / [(x+4)(x-1)(x+3)(x-3)]
Expanding and simplifying both the numerator and denominator:
= [(x^3+5x^2-4x-9)(x+4)(x-1) + (x^3+10x^2+32x+32)(x+3)(x-3)] / [(x+4)(x-1)(x+3)(x-3)]
= [x^5+11x^4+34x^3+20x^2-152x-288] / [(x+4)(x-1)(x+3)(x-3)]
Therefore, the product of the rational expression, when simplified, is:
(x^5+11x^4+34x^3+20x^2-152x-288) / [(x+4)(x-1)(x+3)(x-3)]
To multiply rational expressions and write the product in the simplest form, follow these steps:
Step 1: Factor both the numerator and denominator of each rational expression.
The first rational expression, (x^(2)+2x-3)/(x^(2)+3x-4), can be factored as (x-1)(x+3) / ((x-1)(x+4)).
The second rational expression, (x^(2)+6x+8)/(x^(2)-9), can be factored as (x+2)(x+4) / ((x-3)(x+3)).
Step 2: Write the product of the two expressions.
Multiply the numerators together, (x-1)(x+3)(x+2)(x+4), and multiply the denominators together, (x-1)(x+4)(x-3)(x+3).
Step 3: Simplify the expression.
Look for any common factors in the numerator and denominator that can be canceled out. In this case, we have (x-1) and (x+4) in both the numerator and denominator. Canceling them out, we get:
(x+3)(x+2)(x-3)(x+3) / (x+3)(x-3).
Step 4: Cancel out any remaining common factors.
We have (x+3) and (x-3) in both the numerator and denominator. Canceling them out, we get:
(x+2)(x+3) / 1.
Step 5: Simplify the expression further, if needed.
The final simplified expression is (x+2)(x+3).