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CIE A Level Maths: Pure 3复习笔记1.2.2 Factor & Remainder Theorem
Category:
A-level课程
,
教材笔记
,
福利干货
Date: 2022年8月19日 下午12:58
Factor Theorem
What is the factor theorem?
The factor theorem is a very useful result about polynomials
A
polynomial
is an algebraic expression consisting of a finite number of terms, with non-negative integer indices only
At A level you will most frequently use the factor theorem as a way to simplify the process of factorising polynomials
What do I need to know about the factor theorem?
For a polynomial
f(
x
)
the factor theorem states that:
If
f(
p
) = 0
, then
(
x
-
p
)
is a factor of
f(
x
)
AND
If
(
x
-
p
)
is a factor of
f(
x
)
, then
f(
p
) = 0
Exam Tip
In an exam, the values of
p
you need to find that make
f(
p
) = 0
are going to be integers close to zero.
Try
p
=
1 and -1 first, then 2 and -2, then 3 and -3.
It is very unlikely that you'll have to go beyond that.
Worked Example
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?
Worked Example
Exam Tip
Exam questions will use formal mathematical language which can make factor and remainder theorem questions sound more complicated than they are.
Ensure you are familiar with the various terms from these revision notes
转载自savemyexams
Previous post: CIE A Level Maths: Pure 3复习笔记1.2.1 Polynomial Division
Next post: CIE A Level Maths: Pure 3复习笔记1.2.3 Factorisation
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