Final answer:
The quotient of the polynomial division (5x^4 – 3x^2) by (x+1) is found through synthetic division, yielding a result of 5x^3 - 5x^2 - 2x - 2 with a remainder of 2.
Step-by-step explanation:
To find the quotient of the polynomial (5x^4 – 3x^2) when divided by (x+1), we can use polynomial long division or synthetic division. Since the divisor is linear, let's proceed with synthetic division.
First, identify the coefficients of the polynomial: for 5x^4, the coefficient is 5; for – 3x^2, the coefficient is -3, and include 0s for any missing terms, which in this case would be for x^3 and x. The coefficients are 5, 0, -3, 0, 0 as there is no x term and no constant term.
Next, set up the synthetic division as follows:
- Write down the coefficients: 5, 0, -3, 0, 0.
- Write down the zero of (x+1), which is -1 on the left side of the synthetic division setup.
- Bring down the first coefficient: 5.
- Multiply the zero by the number just written down (5) and write the result under the next coefficient (0).
- Combine these two numbers and continue the process until you reach the end.
We will find that the quotient is 5x^3 - 5x^2 - 2x - 2 with a remainder of 2.
When we divide (5x^4 – 3x^2) by (x+1), we'll use the above process to find the correct quotient which will be the simplified form.
The final answer for the quotient is 5x^3 - 5x^2 - 2x - 2 with a remainder of 2.