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Polynomials and Rational Functions · Paper 1 — Free Maths AA SL Practice Questions

1FoundationSAQ-SPolynomial expressions and their factorizationss5 marksPaper 1~8 min

A rectangular garden has length

(x+3)

metres and width

(x-1)

metres. The area of the garden is

45

square metres.

(a)

Write down an expression for the area of the garden in terms of

x

. [1 mark]

(b)

Show that

x^2 + 2x - 48 = 0

. [2 marks]

(c)

Hence find the value of

x

, justifying why only one value is valid. [2 marks]

Solutions

2FoundationSAQ-SPolynomial expressions and their factorizationss5 marksPaper 1~8 min

Consider the polynomial

P(x) = 2x^3 + 5x^2 - 4x - 3

(a)

Show that

x = 1

is a root of

P(x)

. [2 marks]

(b)

Hence write

P(x)

in the form

(x - 1)(ax^2 + bx + c)

, where

a

b

c \in \mathbb{Z}

. [2 marks]

(c)

Find all real roots of

P(x)

. [1 mark]

Solutions

3MasterySAQ-SPolynomial expressions and their factorizationss6 marksPaper 1~9 min

The volume,

V

cubic metres, of a rectangular box with height

x

metres is given by

V(x) = x(6-x)(8-x)

, where $0 < x < 6$.

(a)

Show that

V(x) = x^3 - 14x^2 + 48x

. [2 marks]

(b)

Hence, find all values of

x

for which the volume is

45

cubic metres. Give your answers in exact form. [4 marks]

Solutions

4ChallengeSAQ-LPolynomial expressions and their factorizationss7 marksPaper 1~11 min

Consider the polynomial function

f(x) = 2x^4 - 5x^3 - 10x^2 + 15x + 18

(a)

(i) Show that

x = 2

is a zero of

f(x)

. [1]
(ii) Hence find a quadratic factor of

f(x)

, showing all division steps. [3 marks]

(b)

Determine the exact values of all real zeros of

f(x)

. [2 marks]

(c)

State the behaviour of

f(x)

x \to \pm\infty

. Hence deduce the number of local turning points of

y = f(x)

, and determine the number of those turning points that are local minima. Justify your answer. [2 marks]

Solutions

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