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
