Multiply and then simplify the rational expression. State any restrictions on the variable.
x^2-7x+10/x^2-4 * x^2+3x+2/x^2-4x-5
how do you get so many simple problems wrong?
(x^2 - 7x + 10) / (x^2 - 4) * (x^2 + 3x + 2) / (x^2 - 4x - 5)
= (x-2)(x-5)/(x-2)(x+2) * (x+2)(x+1)/(x-5)(x+1)
= (x-5)/(x+2) * (x+2)/(x-5)
= 1
You are correct. I apologize for the mistake. The simplified expression is indeed 1.
To clarify, after factoring and canceling out the common factors, we have:
(x - 5)(x - 2)(x + 2)(x + 1) / (x - 2)(x - 5)(x + 2)(x + 1)
After canceling out the common factors, we are left with:
1
Thank you for catching that error.
To multiply the rational expressions, you need to multiply the numerators together and multiply the denominators together. Then, you simplify the resulting expression by canceling out common factors.
Given rational expressions:
(๐ฅ^2-7๐ฅ+10)/(๐ฅ^2-4) * (๐ฅ^2+3๐ฅ+2)/(๐ฅ^2-4๐ฅ-5)
First, let's factor the numerators and denominators of each rational expression:
Numerator 1: ๐ฅ^2-7๐ฅ+10 = (๐ฅ-5)(๐ฅ-2)
Denominator 1: ๐ฅ^2-4 = (๐ฅ-2)(๐ฅ+2)
Numerator 2: ๐ฅ^2+3๐ฅ+2 = (๐ฅ+1)(๐ฅ+2)
Denominator 2: ๐ฅ^2-4๐ฅ-5 = (๐ฅ-5)(๐ฅ+1)
Now, we can rewrite the expression as:
[(๐ฅ-5)(๐ฅ-2)/(๐ฅ-2)(๐ฅ+2)] * [(๐ฅ+1)(๐ฅ+2)/(๐ฅ-5)(๐ฅ+1)]
Next, cancel out the common factors:
[(๐ฅ-5)~(๐ฅ-2)~/(๐ฅ-2)~(๐ฅ+2)] * [(๐ฅ+1)~(๐ฅ+2)/(๐ฅ-5)~(๐ฅ+1)~]
After canceling, the expression simplifies to:
(๐ฅ-5)(๐ฅ+2)/(๐ฅ+2) = ๐ฅ-5
The simplified expression is ๐ฅ-5.
Restrictions on the variable:
To avoid any division by zero, we need to account for any values of ๐ฅ that make the denominators zero.
From the factorization of the denominators, we see that ๐ฅ cannot be equal to 2, -2, 5, or -1. Therefore, the restrictions on the variable are ๐ฅ โ 2, -2, 5, -1.
To multiply two rational expressions, we multiply the numerators together and the denominators together.
The given rational expressions are:
(1) (x^2 - 7x + 10) / (x^2 - 4)
(2) (x^2 + 3x + 2) / (x^2 - 4x - 5)
Let's start by simplifying the numerators and denominators individually.
(1) Numerator of the first expression: (x^2 - 7x + 10)
This can be factored as: (x - 5)(x - 2)
(1) Denominator of the first expression: (x^2 - 4)
This is a difference of squares and can be factored as: (x - 2)(x + 2)
(2) Numerator of the second expression: (x^2 + 3x + 2)
This can be factored as: (x + 2)(x + 1)
(2) Denominator of the second expression: (x^2 - 4x - 5)
This can be factored as: (x - 5)(x + 1)
Now, we can simplify the rational expressions:
(1) Simplified first expression: (x - 5)(x - 2) / ((x - 2)(x + 2))
(2) Simplified second expression: (x + 2)(x + 1) / ((x - 5)(x + 1))
Now, we can multiply the two simplified expressions together:
((x - 5)(x - 2) * (x + 2)(x + 1)) / ((x - 2)(x + 2) * (x - 5)(x + 1))
Next, we can cancel out the common factors in the numerator and denominator:
((x - 5)(x - 2) * (x + 2)(x + 1)) / ((x - 2)(x + 2) * (x - 5)(x + 1))
(x - 2) and (x + 2) cancel out in the numerator and denominator:
(x - 5)(x + 1) / (x - 5)(x + 1)
Finally, we are left with:
1 / 1
Which simplifies to:
1
Restrictions on the variable:
To determine the restrictions on the variable, we need to identify any values that would cause the denominators of the original expressions to become zero.
In this case, the denominators x^2 - 4 and x^2 - 4x - 5 would be zero if:
1) x = 2 or x = -2 (for x^2 - 4)
2) x = 5 or x = -1 (for x^2 - 4x - 5)
Therefore, the restrictions on the variable x are x โ 2, -2, 5, -1.
To multiply and then simplify the rational expression, you can first factor the numerators and denominators:
(x^2 - 7x + 10) / (x^2 - 4) * (x^2 + 3x + 2) / (x^2 - 4x - 5)
The numerator and denominator of the first fraction can be factored:
(x - 5)(x - 2) / (x - 2)(x + 2) * (x + 1)(x + 2) / ((x - 5)(x + 1))
Now we can cancel out the common factors in the numerator and the denominator:
(x - 5)(x - 2)(x + 1)(x + 2) / (x - 2)(x - 5)(x + 1)
Finally, we have:
(x + 2) / 1
The simplified expression is: x + 2
Restrictions on the variable:
The expression is undefined when the denominator is equal to zero. In this case, x cannot equal 2, -2, 5, or -1.