Question

 The polynomial p(x) is defined by
                     p(x) = x3 + ax2 + 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                                                                                                                                      [5]

Answer/Explanation

Ans

2 Substitute x = 2 and equate to zero

    Substitute \(x=-\frac{1}{2}\) and equate to zero

    Obtain \(4a+b+66=0 \ and \ \frac{1}{4}a+b-\frac{21}{4}=0\ or\ equivalents\)

    Solve a relevant pair of linear simultaneous equations
    (Dependent on at least one M mark) 

    Obtain a =− 19  b = 10

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
\(p(x) = x^3 + ax + b\),
where a and b are constants. It is given that (x+2) is a factor of p(x) and that the remainder is 5 when p(x) is divided by (x-3).

(a) Find the values of a and b.
(b) Hence find the exact root of the equation \(p(e^{2y})=0\).

Answer/Explanation

Ans:

  1. Substitute x = -2 and equate to zero
    Substitute x=3 and equate to 5
    Obtain -8-2a+b=0 and 27+3a+b=5 or equivalents
    Solve a pair of relevant linear simultaneous equations for a or b
    Obtain a =-6 and b =-4
  2. Attempt division by x+2 at least as far as \(x^2+kx\)
    Obtain \(x^2-2x-2\)
    Obtain (at least) the positive root \(\frac{2+\sqrt{12}}{2}\) or exact equivalent.
    Equate \(e^{2y}\) to positive root, apply logarithms and use power law
    Obtain \(\frac{1}{2}In(\frac{2+\sqrt{12}}{2})\) or \(\frac{1}{2}In(1+\sqrt{3})\) or exact equivalent

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)

Question

The polynomial p(x )is defined by \(p(x)=ax^{3}+ax^{2}-15x-18\), where a is a constant. It is given that(x − 2 )is a factor of p(x).

(i) Find the value of a.

(ii) Using this value of a, factorise p(x )completely.

(iii) Hence solve the equation \(p(e\sqrt{y})=0,\) giving the answer correct to 2 significant figures.

Answer/Explanation

[2]

4(i) Substitute x = 2 , equate to zero and attempt solution

Obtain a = 4

4(ii) Divide by x − 2 at least as far as the x term 

Obtain\( 4x^{2}+12x+9\)

Conclude\( (x-2 (2x+3)^{2}\)

4(iii) Attempt correct process to solve \(e^{\sqrt{y}} \)= k where k >0

Obtain 0.48 and no others

Question

The polynomial p(x )s defined by \(p(x)=5x^{3}+ax^{2}+bx-16\),

where a and b are constants. It is given that(x − 2) is a factor of p(x) and that the remainder is 27 when p(x )is divided by (x + 1).

(i) Find the values of a and b.

(ii) Hence factorise p(x)completely.

Answer/Explanation

<p(1)Substitute x=2 and equate to zero

Substitute x=-1 and equate to 27

Obtain 4a+2b=-24 and a-b=48 or equivalents 

Solve q relevant pair of simultaneous linear equations 

Obtain a=12 ,b=-36

(ii)

Divide by x-2 at least as far as the term x term to obtain \(5x^{2}+\)(their a+10)x….

Obtain \(5x^{2}+22x+8\)

Obtain (x-2)(5x+2)(x+4)

Question

The polynomial p(x )is defined by\( p(x)=ax^{3}+ax^{2}-15x-18,\)

where a is a constant. It is given that(x − 2 )is a factor of p (x).

(i) Find the value of a.      [2]  

(ii) Using this value of a, factorise p(x )completely . [3]

(iii) Hence solve the equation \(p(e\sqrt{y})=0\) = 0, giving the answer correct to 2 significant figures.

Answer/Explanation

(i) Use quotient rule or equivalent

Obtain correct \(\frac{\frac{5}{x}(2x+1)-10Inx}{(2x+1)^{2}}\) or \(equivalent, or \frac{5}{x}(2x+1)^{-1}-10 Inx (2x+1)^{-2}\) or equivalent

Substitute x =1 to obtain \(\frac{15}{9}or \frac{5}{3}\) or equivalent, www

4(ii) Equate numerator to zero and attempt relevant arrangement

Confirm \(x=\frac{x+0.5}{Inx}\)

4(iii) Use iteration process correctly at least once 

Obtain final answer 3.181

Show sufficient iterations to 6 sf to justify answer or show sign change in interval (3.1805, 3.1815)

Question

(i) Find the quotient when \(4x^{3}+8x^{2}11x+9\) is divided by( 2x + 1), and show that the remainder is 5.                             

(ii) Show that the equation \(4x^{3}+8x^{2}11x+4\) = 0 has exactly one real root

Answer/Explanation

.

4(i) Carry out division at least as far as \(2x^{2}+kx\)

Obtain quotient \(2x^{2}+3x+4\)

Confirm remainder is 5

4(ii) State or imply equation is \((2x+1)(2x^{2}+3x+4)=0\)

Calculate discriminant of 3-term quadratic expression or equivalent

Obtain –23 or equiv and conclude appropriately

Question

The polynomial p(x )is defined by\( p(x)=ax^{3}+bx^{2}-17x-a\)

where a and b are constants. It is given that(x + 2 )is a factor of p(x )and that the remainder is 28 when p
(x )is divided by (x − 2).

(i) Find the values of a and b.

(ii) Hence factorise p(x )completely.

(iii) State the number of roots of the equation \(p(2^{y})=0\),justifying your answer.

Answer/Explanation

(i) Substitute x = –2 and equate to zero

Substitute x = 2 and equate to 28

Obtain -9a+4b+34=0and 7a+4b-62=0 or equivalents

Solve a relevant pair of simultaneous equations for a or b

Obtain a = 6, b = 5 

(ii) Divide by x + 2, or equivalent, at least as far as \(k^{1}x^{2}+k_{2}x\)

Obtain\( 6x^{2}-7x-3\)

Obtain (x+2)(3x+1)(3x-3)

(iii) Refer to, or clearly imply, fact that\( 2^{y}\) is positive  State one

Question

The polynomial (x )is defined by

\(p(x)=8x^{3}+30x^{2}+13x-25\).

(i) Find the quotient when p(x) is divided by (x + 2), and show that the remainder is 5.

(ii) Hence factorise p(x − 5) completely.

(iii) Write down the roots of the equation p(|x| − 5) = 0.

Answer/Explanation

(i) Carry out division, or equivalent, at least as far as \(8x^{2}+kx\)

Obtain correct quotient \(8x^{2}+14x-15\)

Confirm remainder is 5

(ii) State or imply expression is(x+2)(….their quadratic quotient…)

Attempt factorisation of their quadratic quotient

Obtain (x+2)(2x+5)(4x-3)

(iii) State\( \pm \frac{3}{4}\) and no others, following their 3 linear factors

Question

Find the quotient and the remainder when 2x3 + 3x2 + 10 is divided by (x + 2). [3]

Answer/Explanation

Ans:

 Attempt division at least as far as quotient 2x2 x kx 
   Obtain quotient 2x2 x + 2
   Obtain remainder 6 
   Special case: Use of Remainder Theorem to give 6 

Question

 (i) Find the quotient and remainder when

x4 + x3 + 3x2 + 12x + 6

is divided by (x2 − x + 4). [4]

   (ii) It is given that, when

x4 + x3 + 3x2 + px + q

is divided by (x2 − x + 4), the remainder is zero. Find the values of the constants p and q. [2]

   (iii) When p and q have these values, show that there is exactly one real value of x satisfying the equation

x4 + x3 + 3x2 + px + q = 0

and state what that value is. [3]

Answer/Explanation

Ans:

6 (i) Carry out division at least as far as quotient x2 + kx 
          Obtain partial quotient  x2 + 2x
          Obtain quotient x2 + 2x + 1 with no errors seen 
          Obtain remainder 5x + 2 

   (ii) Either Carry out calculation involving 12x + 6 and their remainder ax + b 
                        Obtain p = 7, q = 4 

           Or        Multiply x2 – x + 4 by their three-term quadratic quotient M1
                        Obtain p = 7, q = 4 

   (iii) Show that discriminant of 4x2 – x + 4 is negative 
            Form equation (x2 – x + 4)(x2 + 2x + 1) = 0 and attempt solution M1
            Show that x2 + 2x + 1 = 0 gives one root x = −1 

Question

 The polynomials f(x) and g(x) are defined by

f(x) = x3 + ax2 + b and g(x) = x3 + bx2 − a,

    where a and b are constants. It is given that (x + 2) is a factor of f(x). It is also given that, when g(x) is divided by (x + 1), the remainder is −18.

    (i) Find the values of a and b. [5]

    (ii) When a and b have these values, find the greatest possible value of g(x) − f(x) as x varies. [2]

Answer/Explanation

Ans:

4 (i) Substitute x = −2 in f(x) and equate to zero to obtain −8 +4a + b = 0
          Substitute x = −1 in g(x) and equate to –18 
          Obtain − 1 + 2 – a = – 18 or equivalent 
          Solve a pair of linear equations for a or
          Obtain a = 5 , b = −12 

   (ii) Simplify g(x) − f(x) to obtain form kx2 + c where k < 0 
          Obtain – 17 x2 + 7 and state 7, following their value of c

Question

(i) The polynomial \(x^{3}+ax^{2}+bx+8\), where a and b are constants, is denoted by p(x). It is given that when p(x) is divided by x − 3 the remainder is 14, and that when p(x) is divided by x + 2 the remainder is 24. Find the values of a and b. 
(ii) When a and b have these values, find the quotient when p(x) is divided by \(x^{2}+2x-8\) and hence solve the equation p(x)= 0.

Answer/Explanation

(i) Substitute x = 3 and equate to 14 ( ) 9a + 3b + 35 =14 
Substitute x = −2 and equate to 24 ( ) 4a − 2b = 24 
Obtain a correct equation in any form 
Solve a relevant pair of equations for a or for b 
Obtain a = 1 and b = −10

(ii)Attempt division by \(x^{2}+2x-8\) and reach a partial quotient of x – k 
Obtain quotient x – 1 with no errors seen (can be done by observation) A1
Correct solution method for quadratic e.g. factorisation 
All solutions x = 1, x = 2 and x = –4 given and no others CWO

Question

The polynomial \(ax^{3}-5x^{2}+bx+9\) where a and b are constants, is denoted by p(x). It is given that (2x + 3) is a factor of p(x), and that when p(x) is divided by (x + 1) the remainder is 8.
(i) Find the values of a and b. 
(ii) When a and b have these values, factorise p(x) completely.

Answer/Explanation

i) Substitute \(x=-\frac{3}{2}\) equate to zero
Substitute x = −1 and equate to 8 
Obtain a correct equation in any form A1
Solve a relevant pair of equations for a or for b 
Obtain a = 2 and b = −6

(ii)Attempt either division by 2x + 3 and reach a partial quotient of \(x^{2}+kx\), use of an identity or observation 
Obtain quotient \(x^{2}-4x+3\)

Obtain linear factors x – 1 and x – 3 
[Condone omission of repetition that 2x + 3 is a factor.]

[If linear factors x – 1, x − 3 obtained by remainder theorem or inspection, award B2 + B1]

Question

The polynomial p(x) is defined by \(p(x)=ax^{3}-3x^{2}-5x+a+4\) ,where a is a constant.
(i) Given that (x − 2) is a factor of p(x), find the value of a.
(ii) When a has this value,
(a) Factorise p(x) completely.
(b) find the remainder when p(x) is divided by (x + 1).

Answer/Explanation

(i) Substitute 2 and equate to zero or divide and equate remainder to zero 
Obtain a = 2 

(ii) (a) Attempt to find quadratic factor by division, inspection or identity 
Obtain \(2x^{2}+x-3\)

Conclude (x – 2)(2x + 3)(x – 1) 

(b) Attempt substitution of –1 or attempt complete division by x + 1 
Obtain 6

Question 

The polynomial \(2x^{3}-4x^{2}+ax+b\), where a and b are constants, is denoted by p(x). It is given that when p(x) is divided by (x + 1) the remainder is 4, and that when p(x) is divided by (x − 3) the remainder is 12.
(i) Find the values of a and b. 
(ii) When a and b have these values, find the quotient and remainder when p(x) is divided by \((x^{2}-2)\).

Answer/Explanation

(i) Substitute x = −1, equate to zero and obtain a correct equation in any form 
Substitute x = 3 and equate to 12
Obtain a correct equation in any form 
Solve a relevant pair of equations for a or for b 
Obtain a = −4 and b = 6 A1 
(ii) Attempt division by \(x^{2}-k\) and reach a partial quotient of 2x − k 
Obtain quotient 2x − 4 
Obtain remainder −2

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