A divisor of the form points to the Remainder Theorem: the remainder is the polynomial's value at . A sum of coefficients points to , because every power of equals . If those inputs match, use the remainder directly. The trap is forgetting the leading coefficient when adding the terms.
Hints
- Hint 1
The Remainder Theorem says dividing by leaves . Which divisor gives you the input , where powers of become ?
- Hint 2
At , every power of becomes . Substitute into every term, including . What number appears alongside ?
Step-by-step
Use the remainder at 1
Step 1Turn the remainder into a polynomial value
No Desmos needed. The given remainder already tells you . The Remainder Theorem says the remainder after dividing by is : at , the divisor becomes . So the first remainder gives
- Step 2
Find what the polynomial equals at 1
Put into the given polynomial. Every power of equals , so this adds all the coefficients, including the leading :
- Step 3
Find the requested sum
Set the two expressions for equal: . Subtract the leading : . Compute the difference: . So the requested sum is . Choice C.