The
Q1: a binomial of degree 100:
Β for example: x^100 + 2 (it is called binomial because the number of term is 2: x^100 and 2)
Q2: the coefficients:
Β the main rule is:
if we have Β P(x)= a1X^p + a2X^p-1+ Β + a0 as polynomial, so the coefficients are (a1, a2, ... ao)
i) x3 in 2x + x2 β 5x3 + x4, the coefficient is Β -5
ii) x2 in (Ο/3) x2 + 7x β 3,Β the coefficient isΒ (Ο/3)
iii) x in β3 - 2β2x + 4x2, the coefficient isΒ - 2β2
Q3. If p(y) = 4 + 3y β y2 + 5y3,Β (i) p(0) can be found with
(i) p(0) = 4 + 3*0 β 0Β² + 5*0^3= 4, p(0)= 4, the same method to p(2)
ii) p(2)=4 + 3*2β 2Β² + 5*2^3=49, p(2)=49
iii) p(-1)= 4 + 3*(-1) β (-1)Β² + 5*(-1)^3 = -5, p(-1)=Β -5
Q4. the zero of the polynomial
i) h(x) = 6x-1, to find it, just egalizeΒ h(x) to 0, 6x-1Β =0, it implies 6x=1, and x=1/6, so the zero is x=1/6
ii) p(t) = 2t-3,Β ii) p(t) = 2t-3=0 implies Β 2t=3 and the zero is t=3/2
the remainder in each case if p(x) is divided by g(x)
(i) p(x) = x3 β 6x2 + 2x β 4, g(x) = 1 β 2x
Β Β Β Β Β Β Β x3 β 6x2 + 2x β 4
R(x) = Β Β = (-xΒ²/2 Β + 11x/4 Β + 3/8) Β - 35 / 8(-2x + 1) Β
Β Β Β Β Β Β Β Β Β Β Β 1 β 2x
so the remainder is -Β 35 / 8
(ii) p(x) = 2x3 β 5x2 + 4x β 3, g(x) = 3x + 1.
Β Β Β Β Β Β Β 2x3 β 5x2 + 4x β 3
R(x) = Β = Β 2xΒ²/3 -17x /9 +53/27 - 134 /27(3x+1)
Β Β Β Β Β Β Β Β Β Β Β 3x + 1
the remainder is Β - 134 /27