Question

using factor thereom determine whether g(x) is a factor of p(x) = 2x² + x-2x -1, g(x) = x-3​

Answer 1

Answer:

g(x) is not a factor of p(x)

Note:

★ Remainder theorem : If a polynomial p(x) is divided by (x - a) , then the remainder is given as , R = p(a).

★ Factor theorem : i) If (x - a) is a factor of the polynomial p(x) , then the remainder obtained on dividing p(x) by (x - a) is zero , ie ; R = p(a) = 0.

ii) If the remainder obtained on dividing the polynomial p(x) by (x - a) is zero , ie ; if p(a) = 0 , then (x - a) is a factor of p(x).

Solution:

Here,

p(x) = 2x² + x - 2x - 1

g(x) = x - 3

If g(x) = 0

=> x - 3 = 0

=> x = 3

Now,

Let's find the remainder on dividing p(x) by g(x) using remainder theorem .

Thus,

Remainder , R = p(3)

= 2(3)² + 3 - 2(3) - 1

= 18 + 3 - 6 - 1

= 14 ( R ≠ 0 )

Since the remainder is not equal to zero , ie ; p(3) ≠ 0 , thus according to the factor theorem , g(x) = x - 3 is not factor of p(x) = 2x² + x - 2x - 1 .