Question

Multiply the following rational exp (x⁽²⁾+(-1)x-20)/(x⁽²⁾+(-3)x-18)*(x⁽²⁾+9x+18)/(x⁽²⁾-25)

Answer 1

To multiply the given rational expressions, factor each numerator and denominator, then cancel out any common factors.

Step-by-step explanation:

To multiply the given rational expressions, we need to factor each numerator and denominator and then cancel out any common factors. Let's start with the first expression:

x^2 + (-1)x - 20 = (x + 4)(x - 5)

Now, let's factor the second expression:

x^2 + 9x + 18 = (x + 3)(x + 6)

Next, factor the third expression:

x^2 - 25 = (x + 5)(x - 5)

Finally, factor the fourth expression:

x^2 + (-3)x - 18 = (x - 6)(x + 3)

Now, cancel out the common factors:

((x + 4)(x - 5))/((x - 6)(x + 3)) * ((x + 3)(x + 6))/((x + 5)(x - 5))

The common factors cancel out, leaving us with the simplified multiplication of the rational expressions.

Answer 2

The simplified expression after multiplying the rational expressions is (x + 5)/(x - 5).

Step-by-step explanation:

To multiply the rational expressions, first, factorize each expression's numerator and denominator. Then cancel out common factors to simplify.

The given expression factors into [(x² + (-1)x - 20)/(x² + (-3)x - 18)] * [(x² + 9x + 18)/(x² - 25)].

Factorize each quadratic expression:

(x² + (-1)x - 20) factors into (x - 5)(x + 4).

(x² + (-3)x - 18) factors into (x - 6)(x + 3).

(x² + 9x + 18) factors into (x + 6)(x + 3).

(x² - 25) factors into (x - 5)(x + 5).

Now, substitute the factored expressions into the original expression and simplify:

[(x - 5)(x + 4)/(x - 6)(x + 3)] * [(x + 6)(x + 3)/(x - 5)(x + 5)].

Cancel out the common factors (x - 5) and (x + 3):

[(x + 4)/(x - 6)] * [(x + 6)/(x + 5)].

Further simplify by multiplying the numerators and denominators:

[(x + 4)(x + 6)] / [(x - 6)(x + 5)].

After multiplication, simplify the expression:

(x² + 10x + 24) / (x² - x - 30).

Factorize the numerator and denominator:

(x + 6)(x + 4) / (x - 6)(x + 5).

Finally, cancel out the common factor (x + 6) and (x - 6):

(x + 4)/(x + 5).

This simplified expression, after canceling out common factors, is (x + 4)/(x + 5).