11. Polynomials

Exercises

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Basic exercises
Applied exercises
Challenging exercises

Terms

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The product of a coefficient and a variable is called a term.

The variables that appear in a term stand for things that can receive numerical values. These include things like hourly wage, temperature, car speed, etc. If, for example, a litre of strawberries costs € 2, we can describe the price of strawberries with the term 2x. The term 2x indicates the price formation according to the number x of strawberries. If you buy 4 litres of strawberries, you get a price of € 8 by placing the number 4 in place of the variable x.

Marking terms

The coefficient (number) is written before the variable (letter).
A multiplication sign is not marked in a product containing a number and a variable.
A multiplication sign is not marked in a product containing several variables.
When the coefficient is the number 1, the coefficient is not marked.
The plus or minus signs are marked first.

Example 1

Simplify the following expressions.

a) [[$ 4 \cdot x = 4x $]]
b) [[$ b \cdot 3 = 3b $]]
c) [[$ 1 \cdot y = y $]]
d) [[$ -1 \cdot a = -a $]]
e) [[$ 5 \cdot (-x) \cdot y = -5xy $]]

Example 2

Let’s look which part of the term is a coefficient and which part of the term is a variable.

Polynomials

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When multiple terms are added together, a polynomial is formed. A polynomial with only one term is said to monomial, whereas a polynomial with two terms is a binomial and a polynomial with three terms is a trinomial. The degree of a polynomial means the degree of the term with the greatest exponent.

Understanding the concept of polynomials is the basis for forming and solving equations (mathematical expressions).

Example 3

a) The polynomial [[$ -4x^3+6x-2 $]] is a trinomial and its degree is 3.
b) The polynomial [[$ 2y $]] is a monomial and its degree is 1.
c) The polynomial [[$ 2x^2y -3x $]] is a binomial and its degree is 2.

Example 4

Calculate the value of the trinomial [[$ 2a^3+5b-1 $]], when [[$ a = 4 $]] and [[$ b = -3 $]].

Place the values of the variables a and b into the correct places in the trinomial:

[[$ 2a^3+5b-1 = 2 \cdot 4^3 + 5\cdot (-3) = 2 \cdot 64 - 15 =113 $]]

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