Question

How do you multiply and simplify `\frac { x ^ { 2} + 2x - 3} { x - 1} \cdot \frac { 8x + 56} { x ^ { 2} + 6x + 9}`?

Answer 1

`f(x) = (8(x+7))/(x+3)`

Explanation:

Given:

`f(x) = (x^2+2x-3)/(x-1) * (8x+56)/(x^2+6x+9)`

The surefire way to multiply and simply any set of fractions is to just multiply across the numerators and the denominators.

However, when dealing with polynomial expressions, it is almost certain that factoring them and checking for common factors to divide out will be much easier to accomplish.

Thus, we will factor the polynomial terms:

`f(x) = color(red)(x^2+2x-3)/(x-1) * color(purple)(8x+56)/color(blue)(x^2+6x+9)`

Starting with the first fraction's numerator, by inspection we can factor it:

`color(red)(x^2+2x-3) = color(red)((x-1)(x+3))`

Next, we go to the denominator of the section fraction:

`color(blue)(x^2+6x+9) = color(blue)((x+3)(x+3))`

Third, while not truly necessary, we can factor out the numerator of the second fraction:

`color(purple)(8x+56) = color(purple)(8(x+7))`

From here, we put the factored terms into the expression replacing the original polynomials.

`f(x) = color(red)(x^2+2x-3)/(x-1) * color(purple)(8x+56)/color(blue)(x^2+6x+9) = color(red)((x-1)(x+3))/(x-1)*color(purple)(8(x+7))/(color(blue)((x+3)(x+3)))`

Now we can see which terms divide out:

`f(x)=(color(red)((x-1))color(blue)((x+3)))/(color(red)((x-1)))*color(purple)(8(x+7))/(color(blue)((x+3))(x+3))`

`f(x)=(color(red)(cancel((x-1)))color(blue)(cancel((x+3))))/(color(red)(cancel((x-1))))*color(purple)(8(x+7))/(color(blue)(cancel((x+3)))(x+3))`

`f(x)=(8(x+7))/(x+3)`