Hi. can you please help me with this problem?

When f(x) = x⁵ - 3x⁴ - 6x³ + 3ax² - 24 is divided by x² - 2 the remainder is bx.
Find the values of a and b.
Hence solve the equation f(x) = -8x

thanks!

Forrest,

Since the remainder on dividing f(x) by x² - 2 is bx we know that

f(x) = x⁵ - 3x⁴ - 6x³ + 3ax² - 24 = (x² - 2) g(x) + bx

for some polynomial g(x).

Since (x² - 2) is zero when x = √2 and when x = -√2, substituting x = √2 and x = -√2 in the equation above give two equations in the unknowns a and b. Solve for a and b.

Penny