a polynomial is an algebraic expression that includes addition, subtraction, multiplication, and...
TRANSCRIPT
A polynomial is an algebraic expression that includes addition, subtraction, multiplication, and whole number exponents, such as:
4x3 – 3x2 + 7x + 5
A polynomial can not include roots or fractional
exponents fractions with the variable
on the bottom (or negative
exponents) variables in the exponent
“Polynomial” means “many parts”. The parts of a polynomial
are called terms. Each + or – starts a new
term. 4x3 – 3x2 + 7x + 5
has four terms.
How many terms are in each of these polynomials? x2 – 2x + 3
4n5 + 2n4 + 3n3 – 2n2
5x2y5 + 2x3y6
How many terms are in each of these polynomials? x2 – 2x + 3 3
4n5 + 2n4 + 3n3 – 2n2 4
5x2y5 + 2x3y6 2
In the polynomial
4x3 + 2y2 – 5z + 2 How many terms are there? What is the coefficient on the 1st
term? What is the coefficient on the 3rd
term? What is the base on the 2nd term? Which term is a constant?
In the polynomial
4x3 + 2y2 – 5z + 2 How many terms are there? 4 What is the coefficient on the 1st
term? 4 What is the coefficient on the 3rd
term? -5 What is the base on the 2nd term? y Which term is a constant? 4th
We often classify polynomials by the number of terms they have. In particular …
Monomial Binomial Trinomial
We often classify polynomials by the number of terms they have. In particular …
Monomial = 1 term Binomial = 2 terms Trinomial = 3 terms
Your book will sometimes ask you to write polynomials in standard form. This just means re-arranging the terms so the exponents count down from left to right.
You can have more than one variable in a polynomial.
For instance 5x2y4 + 2xy3 is a binomial where each term has 2 variables.
The degree of a term tells how big that term is.
The degree is the sum of the exponents in a term.
In 5x2y4 + 2xy3, the degree of the 1st term is 6 and the degree of the 2nd term is 4.
What is the degree of each term of this polynomial?
5x2y2 – 2x5 + 4x7y3 – 7x + 5
What is the degree of each term of this polynomial?
5x2y2 – 2x5 + 4x7y3 – 7x + 5 4 5 10 1 0
The degree of a whole polynomial is the largest of the degrees of its terms.
So, the degree of 5x2y4 + 2xy3 is 6.
The degree of 5x2y2 – 2x5 + 4x7y3 – 7x + 5is 10.
What is the degree of 5n5p3 + 2n2p7 – 4n4p3 + 5np6?
What is the degree of 5n5p3 + 2n2p7 – 4n4p3 + 5np6? 8 9 7 7The overall degree is 9.
Special types of polynomials: Quadratic
One variableDegree = 2
CubicOne variableDegree = 3
It’s easy to add or subtract polynomials.
Just combine like terms. When you do that, you
add the coefficients, and leave the variables and coefficients alone.
(5x2 + 3x – 2) + (3x2 – 7x + 8)
= 8x2 – 4x + 6
The variables and exponents act like a label for the terms, which is why they don’t change when you add or subtract.
Simplify:
(6x2y4 + 2x3y + 3x2y2)+ (5x3y – 2x2y4 + 2x2y2)
Simplify:
(6x2y4 + 2x3y + 3x2y2)+ (5x3y – 2x2y4 + 2x2y2)
= 4x2y4 + 7x3y + 5x2y2
Simplify:
(y3 – 2y2 + 3y + 1) + (3y3 + 2y2 – 5y + 2)
Simplify:
(y3 – 2y2 + 3y + 1) + (3y3 + 2y2 – 5y + 2)
4y3 – 2y + 3
Simplify:
(7n4 + 8n3 – 5n2) – (2n4 + 3n3 + n2)
Simplify:
(7n4 + 8n3 – 5n2) – (2n4 + 3n3 + n2)
5n4 + 5n3 – 6n2
Simplify:
(5x2 – 3x + 5) – (2x2 + 4x – 1)
Simplify:
(5x2 – 3x + 5) – (2x2 + 4x – 1)
3x2 – 7x + 6
Simplify:
(7x2 + 2y2) – (4x2 + 2xy – 3y2)
Simplify:
(7x2 + 2y2) – (4x2 + 2xy – 3y2)
3x2 – 2xy + 5y2