Multiply and then simplify the rational expression. State any restrictions on the variable.
x^2-11x+30/x^2-25 * x^2+8x+15/x^2-3x-18
To simplify the expression, we need to factor each of the numerators and denominators and then cancel any common factors.
First, let's factor the numerator and denominator of the first rational expression:
x^2 - 11x + 30 = (x - 5)(x - 6)
x^2 - 25 = (x - 5)(x + 5)
Now, let's factor the numerator and denominator of the second rational expression:
x^2 + 8x + 15 = (x + 3)(x + 5)
x^2 - 3x - 18 = (x - 6)(x + 3)
Multiplying the two rational expressions together, we get:
((x - 5)(x - 6) / (x - 5)(x + 5)) * ((x + 3)(x + 5) / (x - 6)(x + 3))
Next, we cancel out the common factors:
(x - 5) from the first numerator and first denominator,
(x - 6) from the first numerator and second denominator,
(x + 3) from the second numerator and second denominator,
and (x + 5) from the first and second denominator.
After canceling out these common factors, we are left with:
1 / (x + 5)
So, the simplified rational expression is 1 / (x + 5).
Restriction on the variable:
The variable x cannot be equal to -5, as it would make the denominator zero, which is not allowed in a rational expression.
To multiply and simplify the rational expression, we will follow these steps:
Step 1: Factor all the quadratic expressions.
Step 2: Write the product as a single fraction.
Step 3: Simplify the expression by canceling out any common factors.
Step 4: State any restrictions on the variable.
Let's start by factoring each quadratic expression separately:
For the first expression:
x^2 - 11x + 30
= (x - 6)(x - 5)
The factorization of the first expression is (x - 6)(x - 5).
For the second expression:
x^2 + 8x + 15
= (x + 3)(x + 5)
The factorization of the second expression is (x + 3)(x + 5).
Now, we'll write the product as a single fraction:
(x^2 - 11x + 30)(x^2 + 8x + 15)
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(x^2 - 25)(x^2 - 3x - 18)
Multiplying the numerators and denominators, we get:
[(x - 6)(x - 5)(x + 3)(x + 5)]
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(x - 5)(x + 5)(x + 3)(x - 6)
Next, we simplify the expression by canceling out any common factors:
[(x - 6)(x - 5)(x + 3)(x + 5)]
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(x - 5)(x + 5)(x + 3)(x - 6)
The factors (x - 6) and (x + 5) can be canceled out:
= (x - 5)(x + 3)
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(x + 3)(x - 6)
Finally, we're left with the simplified expression: (x - 5)/(x - 6).
Now, let's state any restrictions on the variable.
The expression is a rational expression, so we need to consider any values of x that would make the denominator equal to zero. In this case, the denominator (x - 6) cannot be equal to zero, so we have the restriction: x ≠ 6.
Therefore, the restrictions on the variable x are x ≠ 6.
To multiply rational expressions, you can follow these steps:
1. Factor all the numerator and denominator expressions.
The first rational expression can be factored as:
(x^2 - 11x + 30) = (x - 5)(x - 6)
The second rational expression can be factored as:
(x^2 + 8x + 15) = (x + 3)(x + 5)
The third rational expression can be factored as:
(x^2 - 3x - 18) = (x - 6)(x + 3)
2. Write the factored expressions together as a single fraction.
[ (x - 5)(x - 6) ] / [ (x - 5)(x + 5) ] * [ (x + 3)(x + 5) ] / [ (x - 6)(x + 3) ]
3. Cancel out any common factors between the numerator and denominator.
In this case, we can cancel out (x - 5), (x - 6), and (x + 3), leaving:
[ 1 ] / [ 1 ] * [ 1 ] / [ 1 ]
Therefore, the simplified expression is 1/1, which is equal to 1.
Regarding the restrictions on the variable, when you have a rational expression, you need to exclude any values of the variable that would make the denominator zero. In this case, we have (x - 5), (x + 5), (x - 6), and (x + 3) in the denominator expressions.
The restrictions on the variable are:
1. x ≠ 5 (because (x - 5) would become zero)
2. x ≠ -5 (because (x + 5) would become zero)
3. x ≠ 6 (because (x - 6) would become zero)
4. x ≠ -3 (because (x + 3) would become zero)
Therefore, these values should be excluded from the domain of the original rational expression.