How do you long divide ( (2x^3) - (5x^2) + (4x) - (4) ) div(x - 2) ((2x3)(5x2)+(4x)(4))÷(x2)?

1 Answer
Dec 4, 2016

The remainder is =0=0 and the quotient is =2x^2-x+2=2x2x+2

Explanation:

Let's do the long division

color(white)(aaaa)aaaa2x^3-5x^2+4x-42x35x2+4x4x-2x2

color(white)(aaaa)aaaa2x^3-4x^22x34x2color(white)(aaaaaaaa)aaaaaaaa2x^2-x+22x2x+2

color(white)(aaaaaa)aaaaaa0-x^2+4x0x2+4x

color(white)(aaaaaaaa)aaaaaaaa-x^2+2xx2+2x

color(white)(aaaaaaaaaaa)aaaaaaaaaaa0+2x-40+2x4

color(white)(aaaaaaaaaaaaa)aaaaaaaaaaaaa+2x-4+2x4

color(white)(aaaaaaaaaaaaaa)aaaaaaaaaaaaaa+0-0+00

so, the remainder is 00 and the quotient is 2x^2-x+22x2x+2

You can also use the remainder theorem to see that the remainder is =0=0

If p(x)p(x) is a polynomial, and (x-a)(xa) is a factor of that polynomial

Then, p(a)=(x-a)q(a)+0p(a)=(xa)q(a)+0

q(a)q(a) is the quotient and 00 the remainder

Let f(x)=2x^3-5x^2+4x-4f(x)=2x35x2+4x4

Then, f(2)=2*2^3-5*2^2+4*2-4f(2)=223522+424

=16-20+8-4=24-24=0=1620+84=2424=0

The remainder is =0=0