Question

For which values of a and b, are the zeroes of q(x) = x3+ 2x2 + a also the zeroes
of the polynomial p(x) = x5 - x4- 4x3 + 3x² + 3x + b? Which zeroes of p(x) are
not the zeroes of q(x)?​

Answer 1

Answer:

Step-by-step explanation:

for zeroes of q(x) be also the zeroes of p(x), then q(x) should be the factor of p(x)

that is q(x) should wholly divide p(x) {with remainder 0}

p(x) = x⁵ - x⁴ -4x³ +3x² + 3x + b

q(x) = x³ +2x² +a

p(x) = q(x) . (x² -3x + 2)  + x²(-1-a) +x(3+3a) + b-2a

now remainder to be zero implies

coefficient of x² = 0 ⇒ a = -1

coefficient of x = 0 ⇒ 3a = -3 ⇒ a = -1

and b-2a = 0 ⇒ b = -2

therefore p(x) = x⁵ - x⁴ -4x³ + 3x² - 2

and q(x) = x³+2x²-1

other zeroes of p(x) ;  solving (x² -3x +2 )=0

                x = 1 and x= 2

hope this helps

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