Sunday, December 2, 2018

Remainder Theorem

$\newcommand{\D}{\displaystyle} \newcommand{\qheading}[4]{( \text{CIE, #1, #2, paper #3, question #4})} $

 1. (CIE 2012 summer paper 11 question 2)
 The expression $\D 2x^3+ax^2+bx-30$ is divisible by $\D x+2$ and leaves the remainder of -35 when divided by $\D 2x-1$. Find the values of the constants $\D a$ and $\D b$.[5]

 2. (CIE, 2012, summer, paper 22, question 5)
 It is given that $\D x-2$ is a factor of $\D f(x)=x^3+kx^2-8x-8.$
 (i) Find the value of the integer $\D k$.
 (ii) Using your of $\D k$, find the non-integer roots of the equation $\D f(x)=0$ in the form $\D a\pm \sqrt{b},$ where $\D a$ and $\D b$ are integers. [7]

 3. (CIE 2012, winter, paper 12, question 10)
 A function $\D f$ is such that $\D f(x)=4x^3+4x^2+ax+b.$ It is given that $\D 2x-1$ is a factor of both $\D f(x)$ and $\D f'(x).$
 (i) Show that $\D b = 2$ and find the value of $\D a.$ [5]
Using the values of a and b from part (i),
 (ii) find the remainder when $\D f(x)$ is divided by $\D x + 3,$ [2]
 (iii) express $\D f(x)$ in the form $\D f(x) = (2x - 1)(px^2 + qx + r),$ where p, q and r are integers to be found, [2] (iv) find the values of x for which f(x) = 0. [2]

 4. (CIE 2012, winter, paper 13, question 8)
 (i) Given that $\D 3x^3 + 5x^2 + px + 8 = (x - 2)(ax^2 + bx + c),$ find the value of each of the integers a, b, c and p. [5]
 (ii) Using the values found in part (i), factorise completely $\D 3x^3 + 5x^2 + px + 8.$ [2]

 5. (CIE 2012, winter,paper 21, question 10 )
 (i) The remainder when the expression $\D x^3 + 9x^2 + bx + c$ is divided by $\D x - 2$ is twice the remainder when the expression is divided by $\D x - 1.$ Show that c = 24. [5]
 (ii) Given that $\D x + 8$ is a factor of $x^3 + 9x^2 + bx + 24,$ show that the equation $\D x^3 + 9x^2 + bx + 24 = 0$ has only one real root. [4]

 6. (CIE 2013, summer, paper 12, question 7)
It is given that $\D f(x) = 6x^3 - 5x^2 + ax + b$ has a factor of $\D x + 2$ and leaves a remainder of 27 when divided by $\D x - 1.$
 (i) Show that b = 40 and find the value of a. [4]
 (ii) Show that $\D f(x) = (x + 2)(px^2 + qx + r),$ where p, q and r are integers to be found. [2]
 (iii) Hence solve $\D f(x) = 0.$ [2]

 7. (CIE 2013, summer, paper 21, question 12)
 The function $\D f(x) =x^3+ x^2+ ax+ b$ is divisible by $\D x - 3$ and leaves a remainder of 20 when divided by $\D x + 1.$
 (i) Show that b = 6 and find the value of a. [4]
 (ii) Using your value of a and taking b as 6, find the non-integer roots of the equation $\D f(x) = 0$ in the form $p \pm \sqrt{q},$ where p and q are integers. [5]

 8. (CIE 2013, winter, paper 11, question 6)
 The function $\D f(x)= ax^3+ 4x^2+ bx -2,$ where a and b are constants, is such that $\D 2x - 1$ is a factor. Given that the remainder when $\D f (x)$ is divided by $\D x - 2$ is twice the remainder when $\D f (x)$ is divided by $\D x + 1,$ find the value of a and of b. [6]

 9. $\qheading{2013}{winter}{13}{1}$

 10.$\qheading{2014}{summer }{13}{3}$
 (i) Find, in terms of p, the remainder when $\D x^3 + px^2 + p^2x + 21$ is divided by $\D x + 3.$ [2]
 (ii) Hence find the set of values of p for which this remainder is negative. [3]

 11. $\qheading{2014}{summer}{21}{4}$
 The expression $\D 2x^3+ ax^2+ bx+ 12$ has a factor $x - 4$ and leaves a remainder of -12 when divided by $\D x - 1.$ Find the value of each of the constants a and b. [5]

 12. $\qheading{2014}{summer}{22}{3}$
 (i) Given that $\D x + 1$ is a factor of $\D 3x^3- 14x^2- 7x+ d$, show that d = 10. [1]
 (ii) Show that $\D 3x^3- 14x^2- 7x+ 10$ can be written in the form $\D (x+1)(ax^2+bx+c)$ , where a, b and c are constants to be found. [2]
 (iii) Hence solve the equation $\D 3x^3- 14x^2- 7x+ 10= 0$. [2]

Answers:

$\D
1. a = 5; b = -13
\\ 2. (i)k = 4(ii)-3\pm \sqrt{5}
\\ 3. (i)a = -7(ii)-49\\ (iii)(2x - 1)(2x^2 + 3x - 2)\\ (iv)x = 0:5;-2
\\ 4. (i)p = -26; a = 3; b = 11; c = -4\\ (ii)(x - 2)(x + 4)(3x - 1)
\\ 5. (ii)x^2 + x + 3 = 0 : b^2 - 4ac < 0
\\ 6. (i)a = -14; b = 40\\ (ii)f(x) = (x + 2)(6x^2 - 17x + 20)\\ (iii)x = -2
\\ 7. (i)a = -14; b = 6\\ (ii)-2\pm \sqrt{6}
\\  8. a = -2; b = 2.5
\\ 9. (i)p = 0.5(ii)84
\\ 10. (i)9p - 3p^2 - 6\\ (ii)p < 1; q > 2
\\ 11. a = -3; b = -23
\\ 12. (ii)3x^2 - 17x + 10\\ (iii)-1; 5; 2/3 $

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