Final answer:
The question involves using polynomial long division to divide the given polynomials by the linear factors \(x+5\) and \(x-3\) to express them in the form \(q(x) + \frac{r}{x-a}\), where \(q(x)\) is a quadratic polynomial.
Step-by-step explanation:
To answer the question, we will use polynomial long division to divide each polynomial by the corresponding linear factor, x-a.
(a)
Dividing \(x^3+7x^2+17x+41\) by \(x+5\), we get:
- Divide \(x^3\) by \(x\) to get \(x^2\).
- Multiply \(x+5\) by \(x^2\) and subtract the result from the original polynomial.
- Repeat the process for the resulting polynomial until you have a remainder that is a constant or a linear term.
(b)
Dividing \(2x^3-11x^2+22x-25\) by \(x-3\), we follow a similar process:
- Divide \(2x^3\) by \(x\) to get \(2x^2\).
- Multiply \(x-3\) by \(2x^2\) and subtract the result from the original polynomial.
- Continue dividing until you are left with a remainder that is of lesser degree than the divisor.
The result will be in the form of \(q(x) + \frac{r}{x-a}\) with \(q(x)\) being a quadratic polynomial.