Synthetic division and Remainder Theorem

Question

How do I use the synthetic division and the remainder theorem to determine if x-4 is a factor of 2x^2 + 11x + 12?

Answer 1

x-4 into 2x^2 +11x + 12 = x+19 with remainder 88

it's not a factor if the remainder isn't zero

it's only a factor if the remainder = 0

synthetic division changes the -4 to 4 then copy the coefficients of the terms in the dividend, 2,11, & 12

4 | 2 11 12

2 8

19 12

76

88 is the remainder

2x4 = 8

11+8=19

19x4 =76

12+76 = 88

(2x^2 + 11x +12)/(x-4) = 2x + 19 + 88/(x-4)

let x-4 = 0

x = 4

substitute that value into the quadratic polynomial

2x^2 +11x + 12

2(4)^2 + 11(4) +12

= 32+44 +12 = 88 = the remainder

2x^2 +11x +12

= 2(x^2 +11x/2 + (11/4)^2) + 12 -121/8

= 2(x+11/4)^2 -25/8

vertex = minimum point = (11/4, -25/8) = (2.75, - 3.125)

it has two negative zeros, and no positive zeros

for x-4 to be a factor, 4 would have to be a zero, but there are no positive zeros, using Descartes Rule

2x^2 +11x +12 = 0

x = -11/4 + or - (1/4)sqr(121 -96)

=-11/4 + or - sqr25/4

= -11/4 + or - 5/4

= -16/4 or - 6/4

=-4 or -1.5 = the two negative zeros

2x^2 +11x+12 = 2(x+4)(x+1.5) = (x+4)(2x+3)

factors are (x+4) and (2x+3) or (2x+4) and (x+3/2)

but not (x-4)

-4 |2 11 12

2 -8

3 12

-12

0 = remainder, so x+4 is a factor of 2x^2 +11x +12

2(-4)^2 + 11(-4) +12 = 32 -44 +12 =0