Problems on Remainder theorem

Show that x⁵ – 7x³ – 12x + 18 is divisible by x² + 2x -3

Answer:

Factorizing x² + 2x -3 we get,

= x² + 3x - x -3

= x(x + 3) -1(x +3)

= (x+ 3)(x -1)

Now according to remainder theorem, if F(x) = x⁵ – 7x³ – 12x + 18

Then, F(-3) = 0 and F(1) = 0 for complete divisibility.

Substituting x = -3 we get F(-3) = 0. Similarly for x = 1, we get F(1) = 0.

So, x⁵ – 7x³ – 12x + 18 is divisible by x² + 2x - 3.

If  x² - 2ax + 15 is divisible by x + 5, then find the value of “a”

Answer:

Now according to remainder theorem, if F(x) = x² - 2ax + 15

Then, F(-5) = 0 for complete divisibility.

But F(-5) = 25 + 10a +  15  = 40 + 10a

Since F(-5) = 0, we get 40 + 10a = 0

Or, a = -4