text: maths challenging questions? 1) if f(x) =ax^3 +(a+b)x^2 +(a+2b)x+1 is exactly divisible by (x+1) express b in terms of a and find the quotient when the division is carried out, expressing the coefficeients in terms of a only. find the range of values of a for which the equation f(x) =0 has only one real root.  2) the polynomials p(x) and q(x) are defined by  p(x)=x^8 -1 and q(x) = x^4 +4x^3 +ax^2+bx+5.  (i) show that x-1 and x+1 are factors of p(x). (ii) it is known that when q(x) is divided by x^2-1, a remainder 2x+3 is obtained. find the valuese of a and b. (iii) with these values of a  and b, find the remainder when the polynomaial (3p(x)+4q(x)) is divided by x^2-1 1) plug -1 in for x and simplify.  f(-1) is the remainder.   f(-1) = a(-1)^3 + (a+b)(-1)^2 + (a+2b)(-1) + 1 f(-1) = -a + a + b - a - 2b + 1 f(-1) = -a - b + 1 if it is exactly divisible then the remainder must be 0. -a - b + 1 = 0 or b = 1-a  2) i)show that p(1) and p(-1) equal 0. ii) use long division to obtain (b+4)x + (a+6) b+4 = 2, so b = -2 a+6 = 3, so a = -3 iii) p(x) is divisible by x^2-1 so the remainder is 0. 3(0) + 4(2x+3) = 8x + 12 check my work.