Question

Without actual division prove that the polynomial 2x^3+4x^2+x-34 is exactly divisible by (x-2)

Answer 1

2$x^{3}$ + 4$x^{2}$ +x - 34

= 2x^3 - 4x^2 + 8x^2 +x - 34

= 2x^2(x-2) + 8x^2 + x - 34

=2x^2(x-2) + 8x^2 - 16x +17x - 34

=2x^2(x-2) + 8x(x-2) + 17x - 34

=2x^2(x-2) + 8x(x-2) + 17(x-2)

=(x-2)(2$x^{2}$ + 8x +17)

Therefore polynomial is divisible by x-2

Answer 2

g(x)=x-2
p(x)=2x^3+4x^2+x-34
solution
p(2)=2×(2)^3+4×(2)^2+2-34
=2×8+4×4+2-34
=16+16+2-34
=32+2-34
=34-34
=0
Therefore x-2 is a factor of 2x^3+4x^2+x-34