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 \