It's an introduction to polynomials with an explanation of The Remainder Theorem and The Factor Theorem for Class 10 students. It has some questions for the explanation of the concepts
2. A polynomial looks like this:
INTRODUCTION
A polynomial has:
Variables such as x,y,z with powers as whole
numbers(0,1,2,3…)
Constants like 10
3. 1. Monomial = The polynomial with only one
term. E.g. -2x,5y
2. Binomial = The polynomial with two terms.
E.g. 3x3+5x, 6y+5
3. Trinomial = The polynomial with three terms.
E.g. 4x4+3x2+2, 5y3+4y+9
Types of polynomials
4. When we know the degree we can also give the
polynomial a name:
The Degree of a Polynomial with
one variable is ...
... the largest exponent of that
variable.
DEGREE OF A POLYNOMIAL
6. WAYS TO FIND ZEROES
1. Graphically
Example: 2x+1
2x+1 is a linear polynomial:
The graph cuts the x-axis at -1/2
which means that at this point
the value of the function y=2x+1
is 0
. Basic Algebra
Example: 2x+1
A "root" is when y is zero: 2x+1 = 0
Subtract 1 from both sides: 2x = −1
Divide both sides by 2: x = −1/2
And that is the solution:
x = −1/2
7. Do you remember doing
division in Arithmetic?
"7 divided by 2 equals 3 with a remainder of 1"
Well, we can divide polynomials in a similar manner
Remainder
Theorem
8. POLYNOMIAL DIVISION EXAMPLE
Example: 2x2−5x−1 divided by x−3
f(x) is 2x2−5x−1
d(x) is x−3
After dividing we get the answer 2x+1, but there is a remainder
of 2.
q(x) is 2x+1
r(x) is 2
In the style f(x) = d(x)·q(x) + r(x) we can write:
2x2−5x−1 = (x−3)(2x+1) + 2
You may refer to our another video “Long
Division Method” to see detailed
explanationKeep in
mind
9. The Remainder Theorem
The Remainder Theorem:
When we divide a polynomial f(x) by x−c the remainder is f(c)
When we divide f(x) by the simple polynomial x−c we get:
f(x) = (x−c)·q(x) + r(x)
x−c is degree 1, so r(x) must have degree 0, so it is just some
constant r :
f(x) = (x−c)·q(x) + r
Now see what happens when we have x equal to c:
f(c) =(c−c)·q(c) + r
f(c) =(0)·q(c) + r
f(c) =r
This is the basis of
remainder theorem…
10. Let’s understand with an example
So to find the remainder after dividing by x-c we don't
need to do any division:
Just find f(c)
We didn't need to
do Long Division at
all!
12. The Factor
Theorem
Now ...
We see this when dividing whole numbers. For example 60 ÷ 20
= 3 with no remainder. So 20 must be a factor of 60.
The Factor
Theorem:
When f(c)=0 then x−c
is a factor of f(x)
And the other way
around, too:
When x−c is a factor
of f(x) then f(c)=0