Question 1a

Marks: 2

Below is the graph of a function $f (x) = a x^{3} + b x^{2} + c x + d$ , passing through the points P $(- 3, 0)$ , Q $(- 2, 0)$ , R $(\frac{1}{2}, 0)$ and S $(2, 60) .$ q1a_2-7_ib-aa-hl-maths

a)

Find the values of

a, b, c

and

d .

Key Concepts

Sketching Polynomial Graphs

Question 1b

Marks: 4

The function is translated vertically by the vector $(\begin{matrix} 0 \\ k \end{matrix})$ so that it passes through the point $(3, 190)$ .

b)

Find the value of

k .

Key Concepts

Translations of Graphs

Question 2a

Marks: 2

a)

Given that the equation

2 x^{2} + 4 x - m = 0

has two real solutions, find the set of possible values of

m .

Key Concepts

Discriminants

Question 2b

Marks: 2

b)

Given that the function

f (x) = x^{2} - 5 x + 2 c

has repeated roots, find $c .$

Key Concepts

Discriminants

Question 2c

Marks: 4

c)

Given that the function

g (x) = 2 x^{2} + 2 k x + (\frac{3}{2} - k)

has no real roots, find the set of possible values of

k .

Key Concepts

Discriminants
Quadratic Inequalities

Question 3

Marks: 6

Let a function $f$ be defined by $f (x) = 2 x^{3} + 7 x^{2} - 3 x - 18$ .

(i)

Show that

(x + 3)

is a factor of

f (x)

.

(ii)

Hence factorise

f (x)

fully.

(iii)

Write down all the solutions to

2 x^{3} + 7 x^{2} - 3 x - 18 = 0 .

Key Concepts

Factor Theorem

Question 4a

Marks: 4

a)

Factorise fully

6 x^{3} + x^{2} - 12 x + 5

.

Key Concepts

Factor Theorem
Polynomial Division

Question 4b

Marks: 7

b)

f (x) = a x^{3} + (5 a - 2) x^{2} + (4 a + 2) x - 2 a

(i)

Given that

(x + 3)

is a factor of

f (x)

, find

a .

(ii)

Hence factorise

f (x)

fully.

Key Concepts

Factor Theorem
Polynomial Division

Question 5a

Marks: 2

Consider the polynomial $g (x) = 3 x^{5} - 25 x^{4} + 72 x^{3} - 72 x^{2} - 16 x + 48 .$

a)

Show that 2 is a root of

g (x)

.

Key Concepts

Factor Theorem
Language of Functions

Question 5b

Marks: 5

b)

Given that 2 is a root of

g (x)

with multiplicity 3, factorise

g (x)

fully and hence state the other two roots.

Key Concepts

Polynomial Division
Language of Functions

Question 6

Marks: 5

Consider the function $f (x) = 4 x^{3} + 6 x^{2} - 7 x + 2 .$

(i)

Find the quotient and remainder when

4 x^{3} + 6 x^{2} - 7 x + 2

is divided by

(x - 2) .

(ii)

Hence write

4 x^{3} + 6 x^{2} - 7 x + 2

in the form

(x - 2) (a x^{2} + b x + c) + d,

where

a, b, c

and

d

are constants to be determined.

Key Concepts

Remainder Theorem
Polynomial Division

Question 7a

Marks: 5

The function $f (x) = 2 x^{3} - 5 x^{2} + a x + b$ has $(2 x + 3)$ as a factor, and when $f (x)$ is divided by $(x - 2)$ the remainder is 7.

a)

Show that

a

and

b

must satisfy the simultaneous equations:

$2 a + b = 11$

$3 a - 2 b = - 36$

Key Concepts

Factor Theorem
Remainder Theorem

Question 7b

Marks: 2

b)

Hence find

a

and

b .

Question 8

Marks: 5

Given that $3 + 2 i$ is one of the roots of the equation $x^{3} - 3 x^{2} - 5 x + 39 = 0,$ find the other two roots.

Key Concepts

Complex Roots of Polynomials

Question 9a

Marks: 5

a)

For each of the following polynomials, find the sum of the roots and the product of the roots.

(i)

f (x) = 9 x^{4} + 7 x^{3} - 3 x + 2

(ii)

g (x) = 7 x^{5} - x^{4} + 2 x^{3} + x^{2} - 5 x + 14

(iii)

h (x) = 2 x^{3} - 5 x^{2} - 3 x

(iv)

j (x) = - 3 x^{4} + 2 x^{2} + 5 x - 3

Key Concepts

Sum & Product of Roots

Question 9b

Marks: 5

b)

Consider the equation

6 x^{3} - (4 a) x^{2} - (a + 2) x = 0

.

Given that the sum of the roots is

\frac{8}{3}

, find the three roots of the equation.

Key Concepts

Sum & Product of Roots

Question 10

Marks: 4

For the function $f (x) = a x^{4} + b x^{3} - x^{2} - 24 x - (5 b + 1)$ , the sum of the roots is $\frac{- 7}{2}$ and the product of the roots is $- 18$ . Find the values of $a$ and $b$ .

Key Concepts

Sum & Product of Roots

Question 11a

Marks: 2

The function $f (x) = (x - 3) (x^{2} + 3 x - 4) (a x^{2} + b x + c)$ has three real and two complex roots.

a)

Find the three real roots.

Key Concepts

Solving Quadratic Equations

Question 11b

Marks: 5

It is given for $f (x)$ that the sum of the roots is $- \frac{3}{2}$ and the product of the roots is $- 60$ .

b)

Find the two complex roots, giving your answers in exact form.

Key Concepts

Sum & Product of Roots
Complex Roots of Polynomials

Question 11c

Marks: 4

c)

Given that

f (2) = - 144

, find the values of

a, b

and

c

.

Key Concepts

Solving Polynomial Equations

Question 12a

Marks: 2

$α$ and $β$ are non-real roots of the equation $x^{2} + 3 k x + 2 k + 1 = 0$ , where $k > 0$ is a constant.

a)

Find

α + β

and

α β

, in terms of $k$ .

Key Concepts

Complex Roots of Quadratics
Sum & Product of Roots

Question 12b

Marks: 2

b)

Given that

α^{2} + β^{2} = 3

, show that

(α + β)^{2} = 4 k + 5

.

Key Concepts

Sum & Product of Roots

Question 12c

Marks: 3

c)

Hence find the value of

k

.

Key Concepts

Sum & Product of Roots

Question 13a

Marks: 2

Consider the function $f (x) = k x^{3} + 3 x^{2} + 11 x + 3 k$ , where $k$ is a constant.

It is given that $(2 x - 1)$ is a factor of $f (x)$ .

(a)

Find the value of

k

.

Key Concepts

Factor Theorem

Question 13b

Marks: 3

(b)

Fully factorise

f (x)

.

Key Concepts

Polynomial Division

Question 13c

Marks: 3

(c)

Hence sketch the graph of

y = f (x)

. Clearly label the coordinates of any points where the graph intersects the coordinate axes.

Key Concepts

Sketching Polynomial Graphs
Key Features of Graphs

DP IB Maths: AA HL

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2.7 Polynomial Functions

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