Multiply. State every possible restictions of the variable. Show your work.
3x^2+16x+5/9x^2-1 * 3x^2+2x-1/x^2+4x-5
To multiply the given expression, we need to multiply the numerators and denominators separately and then simplify.
Numerator:
(3x^2 + 16x + 5) * (3x^2 + 2x - 1)
= 9x^4 + 6x^3 - 3x^2 + 48x^3 + 32x^2 - 16x + 15x^2 + 10x - 5
= 9x^4 + 54x^3 + 44x^2 + 26x - 5
Denominator:
(9x^2 - 1) * (x^2 + 4x - 5)
= (3x + 1)(3x - 1)(x + 5)(x - 1)
Therefore, the simplified expression is:
(9x^4 + 54x^3 + 44x^2 + 26x - 5) / (3x + 1)(3x - 1)(x + 5)(x - 1)
To find the possible restrictions of the variable (values that make the denominator equal to zero), we need to set the factors in the denominator equal to zero and solve for x.
Restrictions:
3x + 1 = 0 or 3x - 1 = 0 or x + 5 = 0 or x - 1 = 0
Solving these equations:
3x + 1 = 0
3x = -1
x = -1/3
3x - 1 = 0
3x = 1
x = 1/3
x + 5 = 0
x = -5
x - 1 = 0
x = 1
Therefore, the possible restrictions of the variable are x = -1/3, 1/3, -5, and 1.
To multiply the given expressions, we will follow these steps:
Step 1: Simplify any fractions.
Step 2: Simplify each expression separately.
Step 3: Multiply the simplified expressions.
Step 4: Identify any possible restrictions on the variable(s).
Step 5: Show the final result.
Let's start by simplifying any fractions first.
Given expression:
(3x^2 + 16x + 5) / (9x^2 - 1) * (3x^2 + 2x - 1) / (x^2 + 4x - 5)
Step 1: Simplify fractions (if any)
No fractions are present in the given expression, so we can move on to the next step.
Step 2: Simplify each expression separately.
Expression 1: (3x^2 + 16x + 5) / (9x^2 - 1)
The numerator cannot be factored further, but the denominator is the difference of squares, so we can factor it as follows:
(3x^2 + 16x + 5) / [(3x + 1)(3x - 1)(x + 1)(x - 1)]
Expression 2: (3x^2 + 2x - 1) / (x^2 + 4x - 5)
This expression cannot be factored.
Step 3: Multiply the simplified expressions.
(3x^2 + 16x + 5) / [(3x + 1)(3x - 1)(x + 1)(x - 1)] * (3x^2 + 2x - 1) / (x^2 + 4x - 5)
= [(3x^2 + 16x + 5)(3x^2 + 2x - 1)] / [(3x + 1)(3x - 1)(x + 1)(x - 1)(x^2 + 4x - 5)]
Now, let's move on to identifying possible restrictions on the variable(s).
Restrictions:
The denominator(s) of a fraction cannot be zero. So, we need to check for any values that would make the denominators equal to zero and exclude them from the final result.
From the denominator (9x^2 - 1), we see that x cannot be equal to ±1/3 since it would make the denominator zero. So, x ≠ ±1/3.
From the denominator (x^2 + 4x - 5), we need to find the factors of this quadratic expression:
x^2 + 4x - 5 = (x + 5)(x - 1)
Setting each factor equal to zero:
x + 5 = 0 --> x = -5
x - 1 = 0 --> x = 1
So, x cannot be equal to -5 or 1.
Step 5: Show the final result.
The final result, without the restrictions, is:
[(3x^2 + 16x + 5)(3x^2 + 2x - 1)] / [(3x + 1)(3x - 1)(x + 1)(x - 1)(x^2 + 4x - 5)]
However, considering the restrictions on the variable x, the final result is:
[(3x^2 + 16x + 5)(3x^2 + 2x - 1)] / [(3x + 1)(3x - 1)(x + 1)(x - 1)(x^2 + 4x - 5)], where x ≠ ±1/3, -5, and 1.
To multiply the two expressions, we can follow the basic steps for multiplying rational expressions:
Step 1: Simplify both expressions by factoring if possible.
Step 2: Cancel out any common factors between the expressions.
Step 3: Multiply the numerators together and the denominators together.
Step 4: Simplify the resulting expression, if possible.
Now, let's go through each step in detail.
Step 1: Simplifying both expressions:
3x^2 + 16x + 5 can't be factored further, so it remains the same.
9x^2 - 1 can be factored using the difference of squares formula: (3x)^2 - 1^2
= (3x + 1)(3x - 1)
3x^2 + 2x - 1 can't be factored further, so it remains the same.
x^2 + 4x - 5 can be factored using the quadratic formula: (x + 5)(x - 1)
After factoring, the two expressions become:
(3x^2 + 16x + 5) / (9x^2 - 1) * (3x^2 + 2x - 1) / (x^2 + 4x - 5)
Step 2: Cancel out any common factors between the expressions:
In this case, there are no common factors between the expressions to cancel out.
Step 3: Multiply the numerators together and the denominators together:
(3x^2 + 16x + 5) * (3x^2 + 2x - 1) / ((9x^2 - 1) * (x^2 + 4x - 5))
Expanding the numerator and the denominator:
[(3x^2 * 3x^2) + (3x^2 * 2x) + (3x^2 * -1) + (16x * 3x^2) + (16x * 2x) + (16x * -1) + (5 * 3x^2) + (5 * 2x) + (5 * -1)] /
[(9x^2 * x^2) + (9x^2 * 4x) + (9x^2 * -5) + (-1 * x^2) + (-1 * 4x) + (-1 * -5)]
Simplifying the numerator and the denominator:
(9x^4 + 6x^3 - 3x^2 + 48x^3 + 32x^2 - 16x + 15x^2 + 10x - 5) /
(9x^4 + 36x^3 - 45x^2 - x^2 - 4x + 5)
Combining like terms:
9x^4 + 54x^3 + 44x^2 + 26x - 5 / 9x^4 + 35x^3 - 46x^2 - 4x + 5
Step 4: Simplify the resulting expression, if possible.
At this stage, no further simplification is possible.
Therefore, the final expression after multiplication is:
(9x^4 + 54x^3 + 44x^2 + 26x - 5) / (9x^4 + 35x^3 - 46x^2 - 4x + 5)
Regarding the restrictions on variables, we need to consider any values of x that would make the denominators equal to zero.
In this case, we need to avoid x values that would make the denominators 9x^2 - 1 and x^2 + 4x - 5 equal to zero.
For 9x^2 - 1 = 0:
9x^2 = 1
x^2 = 1/9
x = ±√(1/9) or x = ±1/3
For x^2 + 4x - 5 = 0:
(x + 5)(x - 1) = 0
x = -5 or x = 1
Therefore, the variable x should not take the values -5, 1, -1/3, or 1/3 to avoid division by zero.