Chapter 1 Part B Flashcards
What is a polynomial?
A polynomial is an algebraic expression consisting of a finite number of terms, with non-negative integer indices only
Give the definition of “single-digit prime numbers”
The single-digit prime numbers are 2, 3, 5, and 7.
Definitions:
Prime number: A natural number greater than 1 that has exactly two distinct positive factors: 1 and itself.
Now, for each single-digit prime:
2: The smallest and only even prime number.
3: A prime because its only factors are 1 and 3.
5: A prime because its only factors are 1 and 5.
7: A prime because its only factors are 1 and 7.
The numbers 1, 4, 6, 8, and 9 are not prime because they have more than two factors.
Multiple
Multiple: A number that is the result of multiplying a given number by an integer.
Example: Multiples of 3 are 3, 6, 9, 12, …
Give the definition of the term “Theorem”
Theorem: A mathematical statement that has been proven to be true using logical reasoning and accepted principles.
Example: Pythagoras’ Theorem states that a² + b² = c² in a right-angled triangle.
Give the definition of the term conjecture
Conjecture: A mathematical statement that is believed to be true based on observations but has not yet been proven.
Example: The Goldbach Conjecture states that every even number greater than 2 can be written as the sum of two prime numbers, but it remains unproven.
What does the (∈) symbol mean?
In mathematics, ∈ means “is an element of” and is used in set notation.
For example:
x∈A means “x is an element of set A.”
Factor
Factor: A number that divides another number exactly, without leaving a remainder.
Example: Factors of 12 are 1, 2, 3, 4, 6, 12.
Prime numbers
A natural number greater than 1 that has exactly two factors, 1 and itself.
Example: 2, 3, 5, 7, 11 are prime numbers.
Examples of what are polynomials and non-polynomials
These are polynomials:
- 3x + 5
- 2xy - 4y + 6
- A plain number is a very simple polynomial.
These are NOT polynomials:
- 5x⁻³ (negative exponent)
- Expressions with fractional indices or negative exponents (e.g., 3x⁻¹).
- Terms involving fractional or negative powers are not considered polynomials.
What is polynomial division?
Polynomial division is a method for splitting polynomials into factor pairs (with or without an accompanying remainder term)
What does the Factor theorem state?
The Factor Theorem states that if a polynomial f(x) has a factor (x - a), then f(a) = 0. In other words, if substituting x = a into f(x) gives 0, then (x - a) is a factor of the polynomial. It is useful for finding the factors and roots of polynomials.
Why do you need the factor theorem?
You need to know the Factor Theorem because it helps you simplify and solve polynomials more easily. It allows you to find factors and roots of a polynomial, which is useful for factoring expressions, solving equations, and analyzing functions. It also aids in polynomial division and finding zeros, which are important for understanding the behavior of polynomials and solving related problems in mathematics.
For the polynomial f(x) = x3 + 6x2 - 9x - 14:
a) Use the factor theorem to show that (x - 2) is
a factor of f(.x).
b) Use the factor theorem to find another factor of f(x).
For the polynomial f(x) = x3 + 6x2 - 9x - 14:
a) Use the factor theorem to show that (x - 2) is
a factor of f(.x).
b) Use the factor theorem to find another factor of f(x).
a) To show that (x - 2) is a factor of f(x) using the Factor Theorem:
f(2) = (2)³ + 6(2)² - 9(2) - 14
= 8 + 24 - 18 - 14 = 0
For (x - 2) to be a factor, f(2) must equal 0. Since f(2) = 0, (x - 2) is a factor of f(x).
b) To find another factor:
Try x = -1:
f(-1) = (-1)³ + 6(-1)² - 9(-1) - 14
= -1 + 6 + 9 - 14 = 0
Since f(-1) = 0, by the Factor Theorem, (x + 1) is another factor of f(x).
Explain what is the remainder theorem.
What is the remainder theorem?
The factor theorem is actually a special case of the more general remainder theorem
The remainder theorem states that when the polynomial f(x) is divided by (x - a) the remainder is f(a)
You may see this written formally as f(x) = (x - a)Q(x) + f(a)
In polynomial division
Q(x) would be the result (at the top) of the division (the quotient)
f(a) would be the remainder (at the bottom)
(x - a) is called the divisor
In the case when f(a) = 0, f(x) = (x - a)Q(x) and hence (x - a) is a factor of f(x)– the factor theorem!
How do I solve problems involving the remainder theorem?
If it is the remainder that is of particular interest, the remainder theorem saves the need to carry out polynomial division in full
e.g. The remainder from left parenthesis x squared minus 2 x right parenthesis divided by left parenthesis x minus 3 right parenthesis is 3 squared minus 2 cross times 3 equals 3
This is because if f(x) = x2- 2x and a = 3
If the remainder from a polynomial division is known, the remainder theorem can be used to find unknown coefficients in polynomials
g. The remainder from left parenthesis x squared plus p x right parenthesis divided by left parenthesis x minus 2 right parenthesis is 8 so the value of p can be found by solving 2 squared plus p open parentheses 2 close parentheses equals 8, leading to p space equals space 2
In harder problems there may be more than one unknown in which case simultaneous equations would need setting up and solving
The more general version of remainder theorem is if f(x) is divided by (ax - b) then the remainder is begin mathsize 16px style straight f stretchy left parenthesis b over a stretchy right parenthesis end style
The shortcut is still to evaluate the polynomial at the value of x that makes the divisor (ax - b) zero but it is not necessarily an integer
