FINLANDIA SCHOOLA polynomial is an algebraic expression involving powers of a variable with non-negative integer exponents.
Examples:
The highest power of � is called the degree.
For example:
has degree 4.
8. Remainder Theorem
Rule
If a polynomial � is divided by:
then the remainder is:
In simple terms:
To find the remainder, substitute � into the polynomial.
Worked Example 1
Find the remainder when:
is divided by:
Since the divisor is:
we use:
Therefore:
Therefore, the remainder is 5.
Example 2: Divisor �
Find the remainder when:
is divided by:
Rewrite:
Therefore, substitute:
The remainder is:
9. Factor Theorem
The Factor Theorem is closely connected to the Remainder Theorem.
Rule
If:
then:
In simple language:
If substituting � gives zero, then � is a factor.
Worked Example 3
Show that � is a factor of:
Since the proposed factor is:
we substitute:
Therefore:
10. Factorising a Cubic Polynomial
Consider:
Suppose we test �:
Therefore:
is a factor.
Dividing by �:
gives:
Factorising:
Therefore:
Hence, solving �:
11. Applications in Simplifying Algebraic Expressions
Factor and remainder theorems can help us simplify expressions involving polynomial fractions.
Example
Simplify:
First test �:
Therefore:
is a factor.
Factorise/divide:
by �.
The result is:
Factorise:
Therefore:
Hence:
provided: