Question

HELP PLEASE!!!

Perform the following operations and prove closure.

Answer 1

\[\frac{ x }{ x + 3 } + \frac{ x + 2 }{ x +5 }\]

Answer 2

@Directrix

Answer 3

@ganeshie8

Answer 4

it says "A rational expression is a ratio of two polynomials—a fraction whose numerator and denominator are polynomials. Just as addition, subtraction, and multiplication are closed on integers, they are closed for rational expressions as well, with the addition of division since the division of rational expressions also yields a rational expression."

Answer 5

@Directrix

Answer 6

I believe if you can show that a/b + c/d = (ad + bc) / (bd), then you've proven that this rational expression is closed under addition.

Answer 7

so x/x+3 + x+2/x+5 = (x)(x+5) + (x+3)(x+2)/(x+3)(x+5)

Answer 8

close,

2(x^2 + 5x + 3) / (x+3)(x + 5)

Answer 9

what's the 2?

Answer 10

when you're doing the math to get a common denominator, you end up having to factor out a 2

Answer 11

wait I'm so confused can you do what you did but with the equation thing so I can se better

Answer 12

equation thing?

Answer 13

like where it says at the bottom next to draw

Answer 14

I did cross multiplication and I got \[\frac{ x ^{2} + 5x}{ x ^{2} + 2x + 3x + 6}\]
then, \[\frac{ x ^{2} + 5x}{ x ^{2} + 5x + 6}\]

Answer 15

x/x+3 + x+2/x+5 = (x)(x+5) + (x+3)(x+2) is what you end up with according to the formula

expand (x + 3)(x + 2) out

x^2 + 5x + 6

x^2 + 5x + x^2 + 5x + 6 is what you end up with on top

simplify it

2x^2 + 10x + 6

factor out the 2

2(x^2 + 5x + 3) on top

Answer 16

ooohh ok I see what did wrong.

Answer 17

then what about the bottom?

Answer 18

the bottom just stays (x + 3)(x + 5)

Answer 19

do you have to distribute?

Answer 20

if by distribute you mean expand it, then you would get

x^2 + 8x + 15, which doesn't really do anything, so you would just leave it alone.

Answer 21

ok so the answer is \[\frac{ 2(x ^{2} + 5x + 3) }{ (x + 3)(x + 5) }\]

Answer 22

yup, and since you can write it in this form, then you've shown that this rational expression is closed under addition if x is an integer

Answer 23

and not -3 or -5

Answer 24

ok thanks can you help with three more questions? please.

Answer 25

yeah

Answer 26

ok I'm gonna start a new question

Answer 27

you dont have to