Question
(a) Find the quotient when \(4x^3+17x^2+9x\) is divided by \(x^2+5x+6\), and show that the remainder is 18.
(b) Hence sole the equation \(4x^3+17x^2+9x-18=0\).
Answer/Explanation
Ans:
(a) Carry out division as far as 4x + k
Obtain quotient 4x – 3
Confirm remainder is 18
(b) State or imply equation is \((4x-3)(x^2+5x+6)=0\)
Attempt solution of cubic equation to find three real roots
Obtain \(-3,-2,\frac{3}{4}\)
Question
The polynomial p (x) is defined by
\(p(x)=x^{3}+ax^{2}+bx+16,\)
where a and b are constants. It is given that (x + 2) is a factor of p (x) and that the remainder is 72
when p(x) is divided by (x − 2).
Find the values of a and b
Answer/Explanation
Ans
Substitute x=−2 and equate to zero
Substitute x = 2 and equate to 72
Obtain 4 2 80 a b − += and 4 2 48 0 a b +−= or equivalents
Solve a pair of relevant linear simultaneous equations
Obtain a b = = 5, 14
Question
The polynomial p(x) is defined by
p(x) = ax3 − 11x2 − 19x − a,
where a is a constant. It is given that (x − 3) is a factor of p(x).
(a) Find the value of a. [2]
(b) When a has this value, factorise p(x) completely [3]
(c) Hence find the exact values of y that satisfy the equation p(ey + e−y)= 0. [4]
Answer/Explanation
Ans
7(a) Substitute x = 3 , equate to zero and attempt solution
Obtain a= 6
7 (b) Divide by x− 3 at least as far as the x term
Obtain 2 6 72 x x + + A1
Conclude (x- 3)(3x+ 2)(2x- 1)
7(c) Equate ey + e−y to positive value resulting from part (b)
Multiply by ey and use quadratic formula
Obtain \(e^{y}=\frac{3\pm \sqrt{5}}{2}\)
Obtain \(ln\frac{3\pm \sqrt{5}}{2}\)
Question
The polynomials f (x) and g(x) are defined by
f(x) = 4x3 + ax2 + 8x + 15 and g(x) = x2 + bx + 18,
where a and b are constants.
(a)Given that (x + 3) is a factor of f(x), find the value of a.
(b)Given that the remainder is 40 when g(x) is divided by (x – 2), find the value of b.
(c) When a and b have these values, factorise f(x) – g(x) completely.
Answer/Explanation
(a)Substitute x = −3, equate to zero and attempt solution for a
Obtain a =13
(b)Substitute x=2 , equate to 40 and attempt solution for b
Obtain b = 9
(c)Identify x + 3 as factor of f(x) – g(x)
Attempt, by division or equivalent, to find quadratic factor
Obtain (x + 3) (2x – 1) (2x +1)