Question

HI! PLEASE SHOW FULL SOLUTIONS FOR BOTH QUESTIONS AND ONLY ANSWER IF YOU KNOW! NO CALCULUS PLEASE! THAT WOULD BE VERY APPRECIATED!!

Answer 1

3) p(2) = 22

4) k = 0

Explanation:

Remainder theorem: If the polynomial p(x) is divided by the linear polynomial x -a then the remainder is p(a).

3)p(m) = m⁴ - 2m³ + m² + 12m - 6

p(m) is divided by (m - 2).

m -2 = 0

m = 2. The remainder is p(2)

p(2) = 2⁴ - 2*2³ + 2² + 12*2 - 6

= 16 - 16 + 4 + 24 - 6

= 22

p(2) = 22

4) p(x) = 8x³ + 6x² + x + k

p(x) is divided by (2x + 3).

2x + 3 = 0

2x = -3

`x = (-3)/(2)`

`p((-3)/(2))=-15`

`8*((-3)/(2))^3+6*((-3)/(2))^2-(3)/(2)+k=-15\\\\\\8*(-27)/(8)+6*(9)/(4)-(3)/(2) + k = -15\\\\\\-27 + (27)/(2)-(3)/(2)+k=-15\\\\\\-27 + (24)/(2)+k=-15\\\\`

-27 + 12 + k = -15

k = -15 + 27 - 12

k = 0