Divide the rational expression. Express the quotient as a rational expression in lowest terms.
(x^2+10x+21)/(x^2+11x+28) ÷ (x^2+3x)/(x^2+11x+28)
(i really need to know the steps on how to do this (please just don't give me the answer i already know it)
Rewrite as a product by inverting the denomimator
[(x^2+10x+21)/(x^2+11x+28)] * [(x^2+11x+28)/(x^2+3x)/]
Cancel out the (x^2+11x+28)] terms which appear in both numerator and denominator.
(x^2+10x+21)/(x^2+3x)
Factor numerator and denomimator. Note that (x+3) is a factor of both. Cancel it out.
[(x+7)(x+3)]/[x*(x+3)]
(x+7)/x
THANK YOU SO MUCH!!!! I KEPT TRYING TO DIVIDE EVERYTHING BY THE LCD BUT I JUST KEPT GETTING CONFUSED!
To divide rational expressions, we need to follow these steps:
Step 1: Rewrite the division as multiplication by the reciprocal of the second rational expression.
(x^2+10x+21)/(x^2+11x+28) × [(x^2+11x+28)/(x^2+3x)]
Step 2: Simplify each rational expression separately.
For the first rational expression:
The numerator is a trinomial: x^2+10x+21 can be factored as (x+7)(x+3).
The denominator is also a trinomial: x^2+11x+28 can be factored as (x+7)(x+4).
So, the first rational expression simplifies to:
(x+7)(x+3)/(x+7)(x+4)
For the second rational expression:
The numerator is a trinomial: x^2+11x+28 can be factored as (x+7)(x+4).
The denominator is a polynomial: x^2+3x can be factored as x(x+3).
So, the second rational expression simplifies to:
(x+7)(x+4)/(x)(x+3)
Step 3: Multiply the rational expressions.
[(x+7)(x+3)/(x+7)(x+4)] × [(x+7)(x+4)/(x)(x+3)]
Step 4: Cancel out any common factors in the numerator and the denominator.
The (x+7)(x+4) terms cancel out, leaving only:
1/x
Therefore, the quotient as a rational expression in lowest terms is: 1/x
To divide rational expressions, we follow these steps:
Step 1: Factor the numerator and denominator of both fractions.
We factor each quadratic expression separately. Let's first factor the expressions:
x^2 + 10x + 21 = (x + 3)(x + 7)
x^2 + 11x + 28 = (x + 7)(x + 4)
x^2 + 3x = x(x + 3)
x^2 + 11x + 28 = (x + 7)(x + 4)
Step 2: Flip the second fraction and change the division operation to multiplication.
We change the division operation into multiplication and flip the second fraction:
((x + 3)(x + 7) / (x + 7)(x + 4)) * ((x + 7)(x + 4) / x(x + 3))
Step 3: Cancel out common factors.
We can cancel out the common factors between the numerators and denominators:
((x + 3) * (x + 7)) / (x + 4) * (x + 7) / x
Step 4: Simplify the expression to lowest terms.
Finally, we multiply the numerators together and multiply the denominators together:
= (x + 3)(x + 7)(x + 7) / (x + 4)x
Our simplified expression is:
= (x + 3)(x + 7)^2 / (x + 4)x
So, the quotient expressed in lowest terms is (x + 3)(x + 7)^2 / (x + 4)x.