### 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]

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 $$4x^3+17x^2+9x-18=0$$.

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 $$p(e^{2y})=0$$.

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]

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.

(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.

[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.

<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.

(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

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

.

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.

(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.

(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]

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]

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]

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.

(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.

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).

(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 $$(x^{2}-2)$$.

(ii) Attempt division by $$x^{2}-k$$ and reach a partial quotient of 2x − k