Multiply. State all restictions of the answer. Show your work.
2x^2+7x+3/4x^2-1 * 2x^2+x-1/x^2+2x-3
To simplify the multiplication, we can cancel common factors between the numerator and denominator. However, before we begin, let's identify any restrictions.
1. The expression 4x^2 - 1 cannot equal zero since it would result in a zero denominator.
Solving 4x^2 - 1 = 0:
4x^2 = 1
x^2 = 1/4
x = ±1/2
2. The expression x^2 + 2x - 3 cannot equal zero.
Solving x^2 + 2x - 3 = 0:
Use the quadratic formula or factor:
(x + 3)(x - 1) = 0
x = -3 or x = 1
Therefore, the restrictions of the answer are x ≠ ±1/2 and x ≠ -3.
Now let's perform the multiplication:
(2x^2 + 7x + 3)/(4x^2 - 1) * (2x^2 + x - 1)/(x^2 + 2x - 3)
Simplifying each expression separately:
(2x^2 + 7x + 3)/(2x + 1)(2x - 1) * (2x^2 + x - 1)/(x + 3)(x - 1)
Next, we can cancel common factors:
[(2x + 1)(x + 3)]/[(2x - 1)(x - 1)] * (2x^2 + x - 1)/(x + 3)(x - 1)
Now, multiply the remaining factors:
(2x + 1)(2x^2 + x - 1) = 4x^3 + 2x^2 - 2x + 2x^2 + x - 1 = 4x^3 + 4x^2 - x - 1
(2x - 1)(x + 3) = 2x^2 + 6x - x - 3 = 2x^2 + 5x - 3
Putting it all together:
(4x^3 + 4x^2 - x - 1)/(2x^2 + 5x - 3)
There are no additional restrictions we need to consider in the final answer.
To multiply the given expressions, we'll first simplify each expression and then multiply them together.
Starting with the first expression:
2x^2 + 7x + 3 / (4x^2 - 1)
We notice that the numerator of this expression cannot be further simplified, but the denominator can be expressed as the difference of squares:
4x^2 - 1 = (2x)^2 - 1^2 = (2x + 1)(2x - 1)
So, the first expression can be rewritten as:
(2x^2 + 7x + 3) / ((2x + 1)(2x - 1))
Moving on to the second expression:
2x^2 + x - 1 / (x^2 + 2x - 3)
This can be factored into:
2x^2 + x - 1 = (2x - 1)(x + 1)
So, the second expression becomes:
(2x - 1)(x + 1) / (x^2 + 2x - 3)
Now, we can multiply the two expressions together:
((2x^2 + 7x + 3) / ((2x + 1)(2x - 1))) * ((2x - 1)(x + 1) / (x^2 + 2x - 3))
To simplify, we cancel out common factors:
= (2x^2 + 7x + 3) * (x + 1) / ((2x + 1)(x^2 + 2x - 3))
Multiplying the terms in the numerator:
= (2x^3 + 2x^2 + 7x^2 + 7x + 3x + 3) / ((2x + 1)(x^2 + 2x - 3))
= (2x^3 + 9x^2 + 10x + 3) / ((2x + 1)(x^2 + 2x - 3))
The restrictions of the answer would be any values of x that make the denominator equal to zero:
(2x + 1)(x^2 + 2x - 3) = 0
Setting each factor equal to zero and solving for x:
2x + 1 = 0 => x = -1/2
x^2 + 2x - 3 = 0 => (x + 3)(x - 1) = 0 => x = -3, x = 1
Therefore, the restrictions on the answer are x = -1/2, -3, and 1.
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:
4x^2 - 1 = (2x - 1)(2x + 1)
x^2 + 2x - 3 = (x - 1)(x + 3)
Now, rewrite the expression with the factored denominators:
(2x^2 + 7x + 3) / ((2x - 1)(2x + 1)) * (2x^2 + x - 1) / ((x - 1)(x + 3))
Next, cancel out any common factors between the numerator and the denominator:
(2x + 1) / (2x + 1) = 1
After canceling out the common factors, the expression simplifies to:
(2x^2 + x - 1) / (2x - 1)(x + 3)
Therefore, the answer to the multiplication is:
(2x^2 + x - 1) / (2x - 1)(x + 3)
Restrictions of the answer:
To determine the restrictions, we need to identify any values of x that would make the denominator equal to zero.
From the factored denominators, we have:
(2x - 1)(x + 3)
The restrictions occur when either (2x - 1) or (x + 3) equals zero.
Setting (2x - 1) = 0, we find x = 1/2.
Setting (x + 3) = 0, we find x = -3.
Therefore, the restrictions on the answer are x ≠ 1/2 and x ≠ -3. This means that the value of x cannot be 1/2 or -3.