**Question**

**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*[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**

**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**

**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:**

- 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 - 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**

**Question**

The polynomial p(x) is defined by

p(x) = ax^{3} − 11x^{2} − 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(e^{y} + 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 e^{y} + e^{−y} to positive value resulting from part (**b**)

Multiply by e^{y} and use quadratic formula

Obtain \(e^{y}=\frac{3\pm \sqrt{5}}{2}\)

Obtain \(ln\frac{3\pm \sqrt{5}}{2}\)

**Question**

**Question**

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

f(x) = 4x^{3} + ax^{2} + 8x + 15 and g(x) = x^{2} + 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**

**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**

**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**

**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**

**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**

**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 2*x*^{3} + 3*x*^{2} + 10 is divided by (*x* + 2). [3]

**Answer/Explanation**

Ans:

Attempt division at least as far as quotient 2*x*^{2 }*x* k*x*

Obtain quotient 2*x*^{2 }– *x + 2*

Obtain remainder 6

Special case: Use of Remainder Theorem to give 6

**Question**

** (i) Find the quotient and remainder when**

**x ^{4} + x^{3} + 3x^{2} + 12x + 6**

**is divided by (x ^{2} − x + 4). [4]**

** (ii) It is given that, when**

**x ^{4} + x^{3} + 3x^{2} + px + q**

**is divided by (x ^{2} − 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

x^{4} + x^{3} + 3x^{2} + px + q = 0

and state what that value is. [3]

**Answer/Explanation**

Ans:

**6 (i)** Carry out division at least as far as quotient x^{2} + kx

Obtain partial quotient x^{2} + 2x

Obtain quotient x^{2} + 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 x^{2} – x + 4 by their three-term quadratic quotient M1

Obtain p = 7, q = 4

** (iii)** Show that discriminant of 4x^{2} – x + 4 is negative

Form equation (x^{2} – x + 4)(x^{2} + 2x + 1) = 0 and attempt solution M1

Show that x^{2} + 2x + 1 = 0 gives one root x = −1

**Question**

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

f(x) = x^{3} + ax^{2} + b and g(x) = x^{3} + bx^{2} − 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 *b *

Obtain a = 5 , b = −12

** (ii)** Simplify g(x) − f(x) to obtain form kx^{2} + c where k < 0

Obtain – 17 x^{2} + 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