Aim #1: How do we add and subtract polynomials?
Do Now: Identify how many terms each expression has:
2
a) x + 1
3
b) x y - 3xy + 7y
2
c) 4x + 3x - 2 + x
POLYNOMIAL EXPRESSION: A polynomial expression is either
1. a numerical expression or a variable symbol, (4, x, y, -3)
OR
2
2. the sum of one or more terms.(examples: 4 + y, 4x, -3y, x + 2x + 5)
-1
A polynomial must contain non-negative exponents. (for example, x and
-4
12x are NOT polynomials.
A monomial is a polynomial expression generated using only multiplication
(__×__).
Thus, it does not contain + or – signs.
Examples:
A _______________ is the sum (or difference) of two monomials.
A _______________ is the sum (or difference) of three monomials.
A _______________, as stated above, is the sum of one or more monomials.
Just as the expression (3 + 4) ∙ 5 is a numerical expression but not a number,
(x + 5) + (2x2 – x)(3x + 1) is a polynomial expression, but not technically a polynomial.
We reserve the word polynomial for polynomial expressions that are written
simply as a sum of monomial terms.
§ The degree of a monomial is the sum of the exponents of the variable symbols
that appear in the monomial.
Example:
4x2y3
Degree: ______
• The degree of a polynomial is the degree of the monomial term with the highest
degree.
a) What is the degree of x5 - x3 + 7x?
b) What is the degree of 3 - x2 + 6x8?
Writing polynomials in standard form begin with the highest degree monomial
and continue in descending order.
Rewrite in standard form:
4x2 - 3x3 + 7x - 11 + 2x - 8x4
The leading term of a polynomial is the term of the highest degree that would be
written first if the polynomial is in standard form. The leading coefficient is the
coefficient of the leading term. A constant term is any term with no variables.
Ex:
5 - 2x + 9x2 - 3x3
a) What is the leading term? ________
b) What is the leading coefficient? ________
c) What is the constant term? ________
Adding and Subtracting Polynomials
When adding and subtracting, we combine like terms (variables with the same
degree). When subtracting, don't forget to distribute the negative!
Examples:
2
2
3
(3x - 2x + 3) + (x - 4x - 7)
2
2
a. (4x + x + 7) + (2x + 3x + 1)
b. (3x
3
2
3
2
- x + 8) - (x + 5x + 4x - 7)
c. (3xy + 8x) - 2(xy + 12y)
2
2
d. (5 - t - t ) + (9t + t )
e. (3p + 1) + 6(p - 8) - (p + 2)
2
3
2
f. (w - w + 1) + (w - 2w + 99)
2
2
g. Subtract 2x - 5x + 11 from x + 8x - 20.
5
3
2
(3x - 2x + 3) - (4x - 2x + 9)
4
2
h. Find the sum of 5x - 4x + 7x - 10 and 3x + 2x + 3
i. If you add 2 polynomials together, is the result sure to be another polynomial?
The difference of 2 polynomials?
j. Will the sum of 3 polynomials be another polynomial?
k. Jen writes a polynomial expression using only one variable, x, with degree 4.
Matt writes a polynomial expression using only one variable x, with degree 5.
-What can you determine about the degree of the sum of Jen and Matt‛s
polynomials?
-What can you determine about the degree of the difference of Jen and Matt‛s
polynomials?
l. Suppose Jen writes a polynomial expression using only one variable, x, with
degree of 3 and Max writes a polynomial expression using only one variable, x, with
degree of 3.
-What can you determine about the degree of the sum of Jen and Matt‛s
polynomials?
-What can you determine about the degree of the difference of Jen and Matt‛s
polynomials?
Let's sum it up!
A ___________ is a polynomial expression generated using only multiplication
(__×__). Thus, it does not contain + or – signs. Monomials are written with numerical
factors multiplied together and variable or other symbols each occurring one time
(using exponents to condense multiple instances of the same variable)
A _______________ is the sum (or difference) of monomials.
The __________ of a monomial is the sum of the exponents of the variable
symbols that appear in the monomial.
The degree of a ____________ is the degree of the monomial term with the
highest degree.