Multiply. Show all restictions of the answer. Show your work.
2x^2+7x+3/4x^2-1 * 2x^2+x-1/x^2+2x-3
To multiply the given expressions, we need to multiply the numerators and the denominators separately and then simplify the resulting expression.
Numerator:
(2x^2 + 7x + 3) * (2x^2 + x - 1)
= 4x^4 + 2x^3 - 2x^2 + 7x^3 + 3x^2 - 3x + 3x^2 + x - 1
= 4x^4 + 9x^3 + 6x^2 - 4x - 1
Denominator:
(4x^2 - 1) * (x^2 + 2x - 3)
= (2x + 1)(2x - 1)(x + 3)(x - 1)
= (4x^2 - 1)(x^2 + 2x - 3)
= 4x^4 + 8x^3 - 12x^2 - x^2 - 2x + 3
Final answer:
(4x^4 + 9x^3 + 6x^2 - 4x - 1) / (4x^4 + 8x^3 - 12x^2 - x^2 - 2x + 3)
The restrictions of the answer are the values of x that makes the denominator equal to zero. So, we set the denominator equal to zero and solve for x:
4x^4 + 8x^3 - 12x^2 - x^2 - 2x + 3 = 0
Simplifying the equation, we have:
4x^4 + 8x^3 - 13x^2 - 2x + 3 = 0
Unfortunately, finding the exact values of x that satisfy this equation may require numerical methods or factoring techniques.
Therefore, the restrictions of the answer cannot be determined without further information or additional calculations.
To multiply the given expressions, we need to follow these steps:
Step 1: Simplify any expressions that can be simplified within each expression:
The expression 2x^2 + 7x + 3 can't be simplified further, so we leave it as it is.
The expression 4x^2 - 1 can be factored using difference of squares: (2x + 1)(2x - 1).
The expression 2x^2 + x - 1 can't be simplified further, so we leave it as it is.
The expression x^2 + 2x - 3 can be factored using quadratic factoring: (x + 3)(x - 1).
Step 2: Cancel out any common factors between the numerators and denominators:
The (2x^2 - 1) numerator cancels out with the (x^2 + 2x - 3) denominator, leaving (2x + 1) in the denominator.
The (x^2 + 2x - 3) numerator cancels out with the (4x^2 - 1) denominator, leaving (2x + 1) in the denominator.
Step 3: Multiply the remaining numerators and denominators:
The remaining numerator is (2x^2 + 7x + 3) * (2x^2 + x - 1) = 4x^4 + 9x^3 - 3x^2 + 2x^3 + 4x^2 - x - 2x^2 - 4x + 1 = 4x^4 + 11x^3 + 4x^2 - 5x + 1.
The remaining denominator is (2x + 1) * (2x + 1) = (2x + 1)^2 = 4x^2 + 4x + 1.
Therefore, the final result is (4x^4 + 11x^3 + 4x^2 - 5x + 1) / (4x^2 + 4x + 1).
Restrictions:
The expression has no restrictions since the denominators do not contain any variables that could equal zero.
To multiply the given expressions and show all restrictions, let's follow these steps:
Step 1: Simplify both expressions separately:
- Expressions for multiplication: (2x^2 + 7x + 3) / (4x^2 - 1) * (2x^2 + x - 1) / (x^2 + 2x - 3)
Simplifying the first expression:
- The numerator of the first expression is 2x^2 + 7x + 3.
- The denominator of the first expression is 4x^2 - 1.
Simplifying the second expression:
- The numerator of the second expression is 2x^2 + x - 1.
- The denominator of the second expression is x^2 + 2x - 3.
Step 2: Factorize the denominators and analyze restrictions:
- Factoring the first denominator: 4x^2 - 1 = (2x - 1)(2x + 1)
Here we have two potential restrictions:
1. The denominator cannot be equal to zero: 2x - 1 ≠ 0, which means x ≠ 1/2
2. The denominator cannot be equal to zero: 2x + 1 ≠ 0, which means x ≠ -1/2
- Factoring the second denominator: x^2 + 2x - 3 = (x - 1)(x + 3)
Here we again have two potential restrictions:
1. The denominator cannot be equal to zero: x - 1 ≠ 0, which means x ≠ 1
2. The denominator cannot be equal to zero: x + 3 ≠ 0, which means x ≠ -3
Step 3: Multiply the two simplified expressions:
Multiplying the numerators together and the denominators together gives us:
(2x^2 + 7x + 3)(2x^2 + x - 1) / [(2x - 1)(2x + 1)(x - 1)(x + 3)]
To show work for multiplication, we can use the FOIL method (First, Outer, Inner, Last):
(2x^2 + 7x + 3)(2x^2 + x - 1)
= 2x^2 * 2x^2 + 2x^2 * x - 2x^2 * 1 + 7x * 2x^2 + 7x * x - 7x * 1 + 3 * 2x^2 + 3 * x - 3
= 4x^4 + 2x^3 - 2x^2 + 14x^3 + 7x^2 - 7x + 6x^2 + 3x - 3
= 4x^4 + 16x^3 + 13x^2 - 4x + 3
Therefore, the final answer is:
(4x^4 + 16x^3 + 13x^2 - 4x + 3) / [(2x - 1)(2x + 1)(x - 1)(x + 3)]
Note: The restrictions that we found earlier still apply, so the final answer is valid for all values of x except x = 1/2, -1/2, 1, and -3.