Remainder theorem

Without actual division, 

prove that 2x⁴ -5x³ +2x² –x +2 is divisible by x² – 3x + 2

Answer:

Let, F(x) = 2x⁴ -5x³ +2x² –x +2

And G(x) = x² – 3x + 2

Or, G(x) = (x – 2)(x – 1)

If F(x) is completely divisible by G(x), then( x - 2) and (x - 1) must be a factor of F(x).

Now, as per remainder theorem, F(2) and F(1) should be = 0, if they are completely divisible

So, checking if F(2) and F(1) = 0,

We find F(1) = 2 – 5 + 2 – 1 + 2 = 0

Also F(2) = 32 – 40 + 8 – 2 + 2 = 0

So, F(x) is completely divisible by G(x)

The polynomial x⁴ - 2x³ +3x² –ax + 3a - 7 when divided by x+ 1 leaves the remainder 19. Find the values of  a. 

Answer:

Let: F(x) = x⁴ -2x³ +3x² –ax + 3a – 7

Given,  F(x) – 19 is completely divisible by (x + 1)

Therefore, F(-1) – 19 = 0, as per the remainder theorem.

F(-1) = 4 + 2 + 3 + a + 3a – 7 = 2 + 4a

Therefore, F(-1) – 19 = 2 + 4a – 19 = 0

Or, a = 17/4