math9/15 Recording-2025-09-15T14:37:57.675Z9/15
math 9/15 Polynomials: Definitions, Classification, and Operations
What is a polynomial in this context
An expression built from terms that involve a variable with a whole number exponent.
Today we work with polynomial expressions (not equations or functions yet).
Polynomials should have:
No variable in a denominator.
No variable under a radical.
No fractional exponents on any variable.
If you see a variable in a denominator or under a radical, the expression is not a polynomial (e.g., (\dfrac{1}{x}) or (\sqrt{x})).
We'll later learn how to handle those non-polynomial forms.
Terms and how to identify a polynomial
A polynomial is composed of terms separated by addition or subtraction.
Each term has variables with whole-number exponents (e.g., (x^3), (x^2y), etc.).
Terms are the building blocks of a polynomial; the number of terms helps classify the polynomial (monomial, binomial, trinomial, etc.).
Classification by number of terms
Monomial: a one-term polynomial.
Example: 7x is a monomial.
Binomial: a two-term polynomial.
Example: 9x^2 - 3 is a binomial.
Trinomial: a three-term polynomial.
Example: -6x^3 + 2x^2 - 4 is a trinomial.
Note: These names are especially useful when factoring (e.g., factoring out a greatest common factor or recognizing patterns).
Degree and leading concepts
Degree of a polynomial (when there is a single variable): the highest exponent among its terms.
For terms with multiple variables, the degree of that term is the sum of the exponents of all variables in that term.
Overall degree of a polynomial is the maximum degree among all its terms.
Examples:
For the polynomial
8x^{4} - 3x^{9} + 7x - 4,
the terms have degrees: 4, 9, 1, 0. So the degree is
ext{deg} = 9.The leading term is the term with the highest degree, which is (-3x^{9}).
The leading coefficient (lc) is the coefficient of the leading term: (lc = -3).
The polynomial in decreasing order is: -3x^{9} + 8x^{4} + 7x - 4. (Writing in order helps identify degree and lc quickly.)
Example with multiple variables in a single term:
For the term (-7x^{5}y^{5}), the degree is (5+5 = 10).
For the term (2x^{4}y^{3}), the degree is (4+3 = 7).
For the term (-12xy), the degree is (1+1 = 2).
The polynomial -7x^{5}y^{5} + 2x^{4}y^{3} - 12xy has degree (10) (from the first term), so its degree is 10 and its leading coefficient is (-7).
Ordering polynomials
We usually write terms from highest degree to lowest degree (descending order).
After ordering, the leading term determines the degree and leading coefficient.
Example rearrangement:
Given (-3x^{9} + 8x^{4} + 7x - 4) the leading term is (-3x^{9}) and the lc is (-3).
If a polynomial is not in order, rewrite as descending degrees to identify degree and lc more easily.
Leading term, leading coefficient, and terminology
Leading term: the term with the highest exponent (degree).
Leading coefficient (lc): the coefficient of the leading term.
The leading term leads the polynomial when written in standard form (highest degree first).
These concepts are useful for understanding behavior (e.g., end behavior of graphs) and for factoring strategies.
Degree nuances with multi-variable polynomials
For a term with several variables, add the exponents of all variables in that term to get its degree.
The polynomial degree is the maximum of these term-degrees.
Important reminder: you only add exponents within a single term; you do not add degrees across different terms.
Example: the term (x^{4}y^{3}) has degree (4+3 = 7).
Polynomial operations: overview
We will focus on:
Addition of polynomials: combine like terms only; the variables and their exponents stay the same.
Subtraction of polynomials: equivalent to adding the additive inverse of the second polynomial; still only combine like terms.
Multiplication of polynomials: expand by distributing; both coefficients and exponents can change.
Important rule for addition/subtraction: these operations do not increase the degree of any term; coefficients may change, but exponents do not.
Important rule for multiplication: exponents add when multiplying like bases (the product rule).
Working with parentheses and the +1 / -1 tricks
When a polynomial is placed inside parentheses with a preceding coefficient of +1, you can drop the parentheses without changing the expression. For example:
If you have (+1)(P(x)), you can write just (P(x)).
If a preceding coefficient is -1, you must distribute the negative sign to all terms inside the parentheses.
Example: (-P(x)) is the negation of each term in (P(x)).
These ideas are used frequently when adding or subtracting two polynomials enclosed in parentheses.
Example: adding polynomials with parentheses
Problem setup (as described):
Suppose we have ( (5x^{3} - \tfrac{1}{2}x^{2} + 2x) + (9x^{3} + 10x^{2} - 14x) ).
Process:
Combine like terms by adding coefficients of matching variables/exponents.
For the cubic terms: (5x^{3} + 9x^{3} = 14x^{3}).
For the (x^{2}) terms: (-\tfrac{1}{2}x^{2} + 10x^{2} = \left(9.5\right)x^{2}) (i.e., (\dfrac{19}{2}x^{2})).
For the linear terms: (2x - 14x = -12x).
Result (in standard form):
14x^{3} + \tfrac{19}{2}x^{2} - 12x.
Note: The transcript walked through distribution/combination and emphasized keeping track of coefficients and the fact that exponents do not change under addition/subtraction.
Example: subtracting polynomials with parentheses
Problem setup (as described):
Suppose we have ((13x^{4} - 9x^{2} - 5) - (10x^{4} - 8x^{3} + 3x^{2} - 5)).
Process:
Distribute the negative sign across the second polynomial.
Combine like terms: factor-by-factor.
Key points illustrated in the transcript:
The first parentheses with +1 can be written without the outer parentheses.
The second parentheses with -1 multiplies the terms inside by -1.
After distribution, look for like terms across the two results and combine them.
Notes:
If two terms are exact opposites (e.g., (-7x^{2}) and (+7x^{2})), they cancel, removing that term entirely from the resulting polynomial.
Why we rewrite in standard order
For clarity and for further operations (like factoring) it’s helpful to reorder polynomials in descending degree order: highest degree to lowest degree.
Example transformation: a polynomial might initially be written as
(-7x^{5}y^{5} + 2x^{4}y^{3} - 12xy), but we rewrite as
(-7x^{5}y^{5} + 2x^{4}y^{3} - 12xy) (already in order in this case); if needed, we would arrange to place the highest-degree term first.
Polynomial multiplication: the broad idea
Multiplication employs distribution across terms, unlike addition/subtraction.
When multiplying, both coefficients and exponents change due to the multiplication of terms.
The rules of exponents: for the same base, add exponents: (a^{m} \cdot a^{n} = a^{m+n}).
When multiplying polynomials, you multiply every term in one polynomial by every term in the other (FOIL-like for more terms).
If there are no explicit multiplication signs, adjacent parentheses imply multiplication (an implied multiplication): e.g., ( (P) (Q) ) means multiply P by Q.
Monomial times a polynomial (first kind of multiplication we practice)
Setup: outside a monomial multiplies every term inside the polynomial in parentheses.
Example 1 (consistent with transcript):
Outside: (2x); Inside: (3x^{2} + 4).
Distribution:
(2x \cdot 3x^{2} = 6x^{3}).
(2x \cdot 4 = 8x).
Result: (6x^{3} + 8x).
Example 2 (from transcript style):
Outside: (-5) (or (-5t^{?}) if t is involved); Inside: a polynomial with terms like (4t^{3}, -8t^{2}, 2t).
Distribution results:
(-5 \cdot 4t^{3} = -20t^{3}).
(-5 \cdot (-8t^{2}) = +40t^{2}).
(-5 \cdot (2t) = -10t).
So the product is the sum of these terms: (-20t^{3} + 40t^{2} - 10t).
The key takeaway: for multiplication, exponents add within like bases, and different bases (different variables) stay as separate factors (e.g., (x^{p}y^{q}) from (x^{p} \cdot y^{q})).
Multiplication with multiple variables (same-base rule applies to each base)
If you multiply something like (2xy) by (3x^{2} - y):
Distribute to each term in the second polynomial:
For (3x^{2}): coefficients multiply: (2 \cdot 3 = 6); base components: x^1 * x^2 = x^{3}; y remains because there is no y factor in the second term.
Result term: (6x^{3}y^{1} = 6x^{3}y).
For (-y): (2xy)(-y) = -2x y^{2}) (because x^1 with no x in second term leaves x^1, and y^1 with y^1 yields y^{2}):
Result term: (-2x y^{2}).
Final product: (6x^{3}y - 2xy^{2}).
Important rule: you can only combine like bases across the same base; you combine powers for the same variable, not across different variables.
Example from transcript with two different variables (illustrative):
Multiply (2xy) by (3x^{2} - y) to obtain: (6x^{3}y - 2xy^{2}).
If the second polynomial contains other variables (e.g., (t), etc.), apply the same base-by-base rule for x, y, t, etc., and keep cross-variable factors separate (e.g., a term can be (6x^{3}y t^{2}) if the second factor contains (t^{2}) in combination with the first).
Quick practice notes and tips
Always check whether a term is truly a polynomial term: power must be a nonnegative integer, and no denominator or radical on a base variable.
For addition/subtraction:
Only combine like terms (same variables with same exponents in each term).
Do not change the variables or their exponents by addition/subtraction.
For multiplication:
Distribute to every term in the second polynomial when there is no explicit addition or subtraction sign between two polynomials (implied multiplication).
Use the product rule for exponents: add exponents of like bases.
When using parentheses:
If the outside coefficient is +1, you can drop the parentheses.
If it is -1, distribute the negative sign across all terms inside.
When rewriting polynomials:
Put terms in descending order by degree for easier identification of degree and leading coefficient.
Quick reference of key formulas
Degree of a single-term term:
For a term with multiple variables, (x^{a} y^{b} z^{c}) has degree (a+b+c).
The degree of the polynomial is the maximum of the degrees of its terms:
(\deg(p) = \max
{ \deg(\text{term}i) : \text{term}i \text{ in } p }).
Leading term and leading coefficient (in standard form):
If the polynomial in standard form is
p(x) = a{n}x^{n} + a{n-1}x^{n-1} + \cdots + a{1}x + a{0},
then the leading term is \