What should be added to the polynomial x^2 - 5x + 4, so that x - 1 is a factor of the resulting polynomial? What should be subtracted from the polynomial x^2 - 16x + 30, so that x... What should be added to the polynomial x^2 - 5x + 4, so that x - 1 is a factor of the resulting polynomial? What should be subtracted from the polynomial x^2 - 16x + 30, so that x - 3 is a factor of the resulting polynomial? Find whether or not the first polynomial is a factor of the second polynomial. Read more

Steps to Solve

1. Identify Each Problem Identify which parts of the question you want to tackle. The tasks include finding the difference between the quotient and remainder, and checking for factors in polynomial division.

2. Finding Quotients and Remainders For part (a), apply polynomial long division or synthetic division, recognizing that you want to divide polynomials as specified. The division aims to express $P(x)$ in terms of the divisor $x - 1$.

3. Perform Polynomial Division For example, to find if $f(x) = 2x^2 + x - 3$ is divisible by $g(x) = x - 1$, we use:

   • Synthetic division or long division.
   • Note the remainder directly after division.

4. Identifying the Remainder The remainder from the division gives you the answer to check if $f(x)$ can be fully divided by $g(x)$. If it equals zero, then it's a factor.

5. Subtraction for Polynomial Division In part (c), calculate what needs to be subtracted. For example, if the polynomial is $x^2 - 16x + C$, solving $x^2 - 16x + C - 3x^2 + 10$ can help find the necessary constant $C$.

6. Repeat Divisions For other parts, continue by repeating polynomial long division as necessary to obtain each required solution, carefully observing quotients and remainders.

7. Check for Factors Lastly, for part (e), check if the proposed polynomial factors correctly align with the result derived from earlier division to confirm if they are factors.