Question

Consider the polynomial function g(x) = 8x5 + ax4 + 5x + 20, where a is an unknown real number. If (x+4) is a factor of g(x), what is the value of a?

Answer 1

Your Answer Is 4

f(x) = x⁷ – 6x⁶ + 8x⁵

f'(x) = 7x⁶ - 36x⁵ + 40x⁴

= x⁴ (7x² - 36x + 40) ... (1)

Critical point (x,y): x = 0 , x = (36 ± √(-36)²-4*7*40) / (2*7) = (18±2√11) / 7

x = 0 or x = 3.5 or x = 1.6

plug into (1): y = 0 , y = -394.3 , y = 10.1

extreme : maximum (1.6 , 10.1) minimum (3.5 , -394.3) ... turning points

x⁷ – 6x⁶ + 8x⁵ = x⁵ (x² - 6x + 8) = x⁵ (x-2) (x-4) =0

real zeros: x = 0, x=2, x=4

Answer 2

$g(x) =8x^5 + ax^4 +5x +20 \\\\\text{If}~ x +4 ~ \text{is a factor of g(x) then,}~ \\\\g(-4)=0\\\\\implies 8(-4)^5 +a(-4)^4+5(-4) +20 = 0\\\\\implies 256a -8192 = 0\\\\\implies a = \dfrac{8192}{256} = 32$