2. POLYNOMIAL-
An expression that can have constants, variables and exponents, that
can be combined using addition, subtraction, multiplication and
division, but :
• no division by a variable.
• A variable's exponents can only be 0,1,2,3, etc.
• it can't have an infinite number of terms.
3. A POLYNOMIAL LOOKS LIKE THIS:
example of a polynomial
this one has 3 terms
4. constants (like 3, -20, or ½)
variables (like x and y)
exponents (like the 2 in y2), but only 0, 1, 2, 3, ... etc.
are allowed
Polynomial comes from poly- (meaning "many") and -nominal (in this case meaning "term") ... so
it says "many terms“.
A polynomial can have:
That can be combined using addition, subtraction, multiplication and division.
5. THESE ARE POLYNOMIALS:
3X
X - 2
-6Y2 - (7/9)X
3XYZ + 3XY2Z - 0.1XZ - 200Y + 0.5
512V5+ 99W5
5
(YES, EVEN "5" IS A POLYNOMIAL, ONE TERM IS ALLOWED, AND IT CAN
EVEN BE JUST A CONSTANT!)
AND THESE ARE NOT POLYNOMIALS
3XY-2 IS NOT, BECAUSE THE EXPONENT IS "-2" (EXPONENTS CAN ONLY
BE 0, 1, 2,)
2/(X+2) IS NOT, BECAUSE DIVIDING BY A VARIABLE IS NOT ALLOWED
1/X IS NOT EITHER
√X IS NOT, BECAUSE THE EXPONENT IS "½" (SEE FRACTIONAL
EXPONENTS)
BUT THESE ARE ALLOWED:
X/2 IS ALLOWED, BECAUSE YOU CAN DIVIDE BY A CONSTANT
ALSO 3X/8 FOR THE SAME REASON
6. Monomial, Binomial, Trinomial
There are special names for polynomials with 1, 2 or 3 terms:
There is also quadrinomial (4 terms) and quintinomial (5 terms),
but those names are not often used.
Can Have Lots and Lots of Terms
Polynomials can have as many terms as needed, but not an infinite number of terms.
7. VARIABLES
Polynomials can have no variable at all
Example: 21 is a polynomial. It has just one term, which is a
constant.
Or one variable
Example: x4-2x2+x has three terms, but only one variable
(x)
Or two or more variables
Example: xy4-5x2z has two terms, and three variables (x, y
and z)
Example: xy4-5x2z has two terms, and three variables (x, y
and z)
8. WHAT IS SPECIAL ABOUT
POLYNOMIALS?
Because of the strict definition, polynomials are easy to
work with.
For example we know that:
•If you add polynomials you get a polynomial
•If you multiply polynomials you get a polynomial
So you can do lots of additions and multiplications, and
still have a polynomial as the result.
Also, polynomials of one variable are easy to graph, as
they have smooth and continuous lines.
See how nice and
smooth the curve is?
Example: x4-2x2+x
9. DEGREE
The degree of a polynomial with only one variable is the
largest exponent of that variable.
The Degree is 3 (the
largest exponent of x)
Example:
10. STANDARD FORM
The Standard Form for writing a polynomial is to put the
terms with the highest degree first.
Example: Put this in Standard Form: 3x2 - 7 + 4x3 + x6
The highest degree is 6, so that goes first, then 3, 2 and then the constant
last:
x6 + 4x3 + 3x2 - 7
The highest degree is 6, so that goes first, then 3, 2 and then the constant last:
x6+ 4x3 + 3x2 – 7
11. Adding and Subtracting Polynomials
To add polynomials we simply add any like terms together .. so what is a like term?
Like Terms
Like Terms are terms whose variables (and their exponents such as the 2 in x2)
are the same.
In other words, terms that are "like" each other.
Note: the coefficients (the numbers you multiply by, such as "5" in 5x) can be
different.
7x x -2x πx
(1/3)x
y2
-2xy2 6xy2 xy2/2
Example:
are all like terms because the variables are all x
Example:
are all like terms because the variables are all xy2
12. DEGREE OF POLYNOMIALS
Linear polynomial- A polynomial of degree one is called a linear
polynomial.
EXAMPLES- 2x – 1,√2 y + 1, 2 – u
Quadratic polynomial- A polynomial of degree two is called a
quadratic polynomial.
EXAMPLES- 5 – 𝑦2, 4y + 5𝑦2 and 6 – y – 𝑦2
Cubic polynomial- A polynomial of degree three is called a cubic
polynomial.
13. WHAT IS ZERO OF POLYNOMIAL
It is a solution to the polynomial equation, P(x) = 0.
It is that value of x that makes the polynomial equal to
0.
In other words, the number r is a root of a polynomial
P(x)
if and only if P(r) = 0.
Example - Let P(x) = 5x3 − 4x2 + 7x − 8. Then a root of that
polynomial is 1 because, according to the definition:
P(1) = 5· 13 − 4· 12 + 7· 1 − 8
= 5 − 4 + 7 − 8
= 0.
It is traditional to speak of a root of a polynomial. Of a
function in general, we speak of a zero.
14. REMAINDER THEOREM
Let p(x) be any polynomial of degree greater than or equal to one and let a be any
real number. If p(x) is divided by the linear polynomial x – a, then the remainder is
p(a).
Proof : Let p(x) be any polynomial with degree greater than or equal to 1.
Suppose that when p(x) is divided by x – a, the quotient is q(x) and the
remainder is r(x), i.e., p(x) = (x – a) q(x) + r(x)
Since the degree of x – a is 1 and the degree of r(x) is less than the degree of x – a,
the degree of r(x) = 0. This means that r(x) is a constant, say r.
So, for every value of x, r(x) = r.
Therefore,p(x) = (x – a) q(x) + r
In particular, if x = a, this equation gives us p(a) = (a – a) q(a) + r = r,
which proves the theorem.
15. HOW WE CAN DIVIDE BY ANY NON-
ZERO POLYNOMIAL.
16. ALGEBRAIC IDENTITIES
• (a + b)2 = a2 + 2ab + b2
• (a – b)2 = a2 – 2ab + b2
• a2 – b2 = (a + b)(a – b)
• (x + a)(x + b) = x2 + (a + b)x + ab
• (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
• (a + b)3 = a3 + b3 + 3ab (a + b)
• (a – b)3 = a3 – b3 – 3ab (a – b)
• a3 + b3 = (a + b)(a2 – ab + b2)
• a3 – b3 = (a – b)(a2 + ab + b2)
• a3 + b3 + c3 – 3abc = (a + b + c)(a2 + b2 + c2 – ab – bc
– ca)
