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iGCSE (2021 Edition)

5.01 Polynomials

Worksheet
Definition of a polynomial
1

Determine whether each of the following is a polynomial:

a

2 x^{3} - 4 x^{5} + 3

b

\sqrt{2} x^{6} + 7 x

c

A\left( x \right) = 4 x^{\frac{1}{4}} + 2 x^{5} + 2

d

3 x^{3} + \dfrac{2}{x^{7}} - 1

e

\left(x^{3} + 3\right) \left(\sqrt{7} + 2 x^{ - 2 }\right)

2

State the definition for a polynomial in terms of x.

3

State whether each of the following is a multivariable polynomial:

a
4^{x} + u^{4} - 9 u x
b
\dfrac{1}{9} z^{2} + 6 z y - 3 y^{4} - 3
c
t^{3} - 4 t^{2} + 4 t - 8
d
y^{4} - \dfrac{3}{2} x + 9 x^{4} y - 4 z^{3} + 4 x^{\frac{1}{2}}
Parts of a polynomial
4

Consider the expression - 2 a^{3}.

a

Is this expression a monomial?

b

For this expression state:

i

The variable

ii

The coefficient

iii

The degree

5

Consider the term 9 x^{8}.

a

What is the coefficient of 9 x^{8}?

b

What is the exponent?

6

For each the following expressions:

i
State how many terms there are.
ii
Write down the coefficients of the terms.
a

5 y^{3}

b

- 5 y^{9} - y

c

x + 7 x^{6} - x^{9}

7

For the following polynomials, state:

i
The degree
ii
The leading coefficient
iii
The constant term
a

P \left( x \right) = \dfrac{x^{7}}{5} + \dfrac{x^{6}}{6} + 5

b

P \left( x \right) = 2 x^{7} + 2 x^{5} + 2 x + 2

c

P \left(x\right) = 7 \sqrt{6} - \sqrt{5} x^{5} + 5

d

P(x)=6 - \dfrac{7}{10} x^{3}

8

For the following polynomials:

i
State the degree of the polynomial.
ii
State whether the polynomial is a monomial or a binomial.
a

2 x^{4} y^{7}

b

7 x^{5} y + 8 x y^{3}

c

5 x + 0.6

d

7 x^{2} + 2 x - 3

e

- 2^{7} x y^{4}

9

State the degree of the following terms:

a

8

b

- 4 y^{3}

c

- 4^{8} y^{6}

10

State the numerical coefficient of:

a
3 k^{6}
b
- k
c
\dfrac{t}{3}
d
\dfrac{6 m}{7}
11

True or False: The degree of a polynomial in x is the largest coefficient of any of the terms of the polynomial.

12

For the polynomial P \left( x \right) = 8:

a

What is the degree?

b

What is the constant term?

13

Suppose we are given a polynomial of degree n. What is the name of the process where we find two polynomials of degree less than n that multiply together to give the original?

14

What operation is the opposite of factorising?

15

Consider the expression: 12 + 7 p^{2}. What is the coefficient of p?

16

Which terms are like terms with 5 in the following expression?

5 + 5 x^{2} + 13 + \frac{x^{2}}{5} + \frac{1}{5} + 2 x

Application
17

From 2006 to 2016, the population of Australia, measured in millions, could be estimated by the polynomial 0.02 x^{2} + 0.06 x + 20.7, where x is the number of years since 2006.

a

Use the polynomial to estimate the population of Australia (in millions) in 2006. Round your answer to one decimal place.

b

Use the polynomial to estimate the population of Australia (in millions) in 2012. Round your answer to two decimal places.

c

According to the estimation given by the polynomial, how much (in millions) did the population of Australia increase from 2006 to 2012? Round your answer to two decimal places.

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