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 q(x) are not the zeroes of p(x)? ANSWER IT ASAP

Answer 1

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

Answer 2

since,q(x) is factor of p(x)
Dividing q(x) with p(x)

By long division method 〰refer〰attachment.〰

Remainder = 0
-(1+a)x²+(3+3a)x+(b-2a) = 0

comparing coefficient of x²
→ -(1+a) = 0
→ -1-a = 0
→(((((( a = -1 ))))))

comparing coefficient of constant terms

→(b-2a) = 0
→[b-2(-1)] = 0
→((((((( b = -2 ))))))

a=(-1) @nd b=(-2) are common zeroes of p(x) and q(x)

now,
the zeroes of p(x) that are not the zeroes of q(x)

solving x²-3x+2. (Quotient)

→x²-3x+2
→x²-2x-x+2
→(x-2)(x-1)

x=2,1 are not zeroes of q(x)

hope it helps______with reGarDs ===== SnehaG☑