Multiply. State any restictions of the variable. Show your work.
2x^2+7x+3/4x^2-1 * 2x^2+x-1/x^2+2x-3
To multiply these fractions, we need to factor the numerators and denominators and then simplify where possible.
Given:
(2x^2 + 7x + 3) / (4x^2 - 1) * (2x^2 + x - 1) / (x^2 + 2x - 3)
First, let's factor each expression:
2x^2 + 7x + 3 = (2x + 3)(x + 1)
4x^2 - 1 = (2x - 1)(2x + 1)
2x^2 + x - 1 = (2x - 1)(x + 1)
x^2 + 2x - 3 = (x + 3)(x - 1)
Now, we can plug in these factors into the original expression:
((2x + 3)(x + 1)) / ((2x - 1)(2x + 1)) * ((2x - 1)(x + 1)) / ((x + 3)(x - 1))
Notice that (2x - 1)(x + 1) appears in both the numerator and denominator, so it can be canceled out:
(2x + 3) / ((2x - 1)(x + 3))
Therefore, the final simplified expression is:
(2x + 3) / ((2x - 1)(x + 3))
Restrictions of the variable:
The restrictions occur when the denominators are zero since division by zero is undefined. In this case, we need to check where the denominators (2x - 1) and (x + 3) equal zero:
(2x - 1) = 0
2x = 1
x = 1/2
(x + 3) = 0
x = -3
So, the restrictions of the variable are x ≠ 1/2 and x ≠ -3.
To multiply the given expression, let's first factorize the expressions:
2x^2+7x+3 = (2x+1)(x+3)
4x^2-1 = (2x+1)(2x-1)
2x^2+x-1 = (2x-1)(x+1)
x^2+2x-3 = (x-1)(x+3)
Factoring the expressions helps us cancel out common factors.
Now, let's combine the expression and simplify:
(2x^2+7x+3)/(4x^2-1) * (2x^2+x-1)/(x^2+2x-3)
[(2x+1)(x+3)]/[(2x+1)(2x-1)] * [(2x-1)(x+1)]/[(x-1)(x+3)]
Now, cancel out common factors:
[(2x+1) * (x+3) * (2x-1) * (x+1)] / [(2x+1) * (2x-1) * (x-1) * (x+3)]
The (2x+1), (2x-1), and (x+3) terms in the numerator and denominator cancel out, leaving us with:
(x+1) / (x-1)
So, after canceling out common factors, the simplified expression is:
(x+1) / (x-1)
There are no restrictions mentioned for the variable x, so the answer is valid for all values of x except for x=1 where the denominator becomes zero and the expression is undefined.
To multiply the given expression:
(2x^2 + 7x + 3) / (4x^2 - 1) * (2x^2 + x - 1) / (x^2 + 2x - 3)
First, let's factorize the denominators to determine any restrictions on the variable x:
4x^2 - 1 = (2x + 1)(2x - 1)
x^2 + 2x - 3 = (x + 3)(x - 1)
From these factorizations, we can see that the expressions are defined for all real values of x except when the denominators equal zero, which means:
2x + 1 = 0 → x = -1/2
2x - 1 = 0 → x = 1/2
x + 3 = 0 → x = -3
x - 1 = 0 → x = 1
These values of x are the restrictions we need to keep in mind while solving the given expression.
Now, let's multiply the numerators and denominators:
(2x^2 + 7x + 3) * (2x^2 + x - 1) / ((4x^2 - 1) * (x^2 + 2x - 3))
Expanding the numerators:
(2x^2 + 7x + 3) * (2x^2 + x - 1) = 4x^4 + 2x^3 - 2x^2 + 14x^3 + 7x^2 - 7x + 6x^2 + 3x - 3
= 4x^4 + 16x^3 + 21x^2 + 6x - 3
Expanding the denominators:
(4x^2 - 1) * (x^2 + 2x - 3) = (2x + 1)(2x - 1)(x + 3)(x - 1)
= (2x + 1)(2x - 1)(x + 3)(x - 1)
Therefore, the simplified expression is:
(4x^4 + 16x^3 + 21x^2 + 6x - 3) / ((2x + 1)(2x - 1)(x + 3)(x - 1))
Remember to consider the restrictions on x we found earlier, namely x = -1/2, 1/2, -3, and 1.