Math 1500 Day 1

Fractions

• Key terms
  • Numerator: the top part of a fraction
  • Denominator: the bottom part of a fraction
  • Reciprocal: the reciprocal of a nonzero number a is $\frac{1}{a}$
• Reduction and simplest form
  • Reduce by dividing numerator and denominator by their greatest common divisor (g): $\frac{a}{b} = \frac{a/g}{b/g}$
• Common denominator
  • If adding fractions with different denominators, rewrite so both have the same denominator
• Adding fractions
  • For $\frac{a}{b} + \frac{c}{d}$ with $b \neq 0, d \neq 0$, the sum is:
    $\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$
  • Example: $\frac{1}{2} + \frac{1}{3} = \frac{1\cdot 3 + 1\cdot 2}{2\cdot 3} = \frac{5}{6}$
• Subtracting fractions
  • For $\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}$
• Practical rewrite
  • Example: Align denominators; $\frac{3}{4} + \frac{5}{6}$
    → rewrite to common denominator 12: $\frac{3}{4} = \frac{9}{12}, \frac{5}{6} = \frac{10}{12}$
    → sum = $\frac{19}{12}$
• Denominator restrictions
  • Denominators cannot be zero

Polynomials (Basics)

• What is a polynomial?
  • A polynomial is a sum of terms; each term is a coefficient times a product of variables with nonnegative integer exponents
  • A term can be written as $a x^{n} y^{m} \cdots$ where a is the coefficient, x, y are variables, and n, m are exponents
• Exponents and bases
  • For any base b and natural numbers n, m: $b^{n} \cdot b^{m} = b^{n+m}$
  • Distinct bases do not combine: $x^{n} \cdot y^{m} \neq x^{n+m} y^{m}$
  • Zero exponent: $b^{0} = 1\quad (b \neq 0)$
  • Example: $2^{3} = 8$
• Terms, coefficients, and variables
  • A polynomial is a sum of terms
  • A term has a coefficient and a variable part; the degree of a term is the sum of its exponents
• Monomial, Binomial, Trinomial
  • Monomial: one term
  • Binomial: two terms
  • Trinomial: three terms
• Polynomial in one or more variables
  • Example in one variable: $2x^{2} - 5x + 7$
  • Example in multiple variables: $5x^{2}y^{2} + 6x^{2} + 3xy$
• Degree of a polynomial and of a term
  • Degree of a monomial is the sum of exponents in that term
  • Degree of a polynomial is the highest degree among its terms
• Multivariable degree
  • The degree of a term like $5x^{3}y^{2}$ is $3 + 2 = 5$
  • The degree of a term like $6x^{2}y^{0}$ is $2 + 0 = 2$
  • The degree of a term like $3xy$ is $1 + 1 = 2$
• Examples from notes
  • The polynomial $5x^{3}y^{2} + 6x^{2} + 3xy + 1$ has degree $ext{deg} = \max{5, 2, 2, 0} = 5$
• Terminology recap
  • Monomial: 1 term
  • Binomial: 2 terms
  • Trinomial: 3 terms
  • Constant term: degree 0 (no variables)
  • Leading term: the term with the highest degree (context-dependent)

Exponent rules (summary)

• Product rule
  • $x^{n} \cdot x^{m} = x^{n+m}$
• Zero exponent
  • $x^{0} = 1$
• Distinct bases
  • You cannot combine exponents across different bases

Degree and terms in detail

• Degree of a polynomial: the highest total degree among its terms
• Degree of a monomial: the sum of exponents in that term
• Illustrative examples
  • For the term $2$, viewed as $2x^{0}$, the degree is $0$
  • For the term $5x^{3} - 2x^{2} + 1$, the highest degree is $3$
• Multivariable polynomials can have more than one variable

Addition and subtraction of polynomials

• Like terms
  • Like terms have the same variable part (same exponents for each variable)
• Adding polynomials by combining like terms
  • Example:
  • $(8x^{2} + x + 3) + (3x^{2} - 2x - 1)$
  • Combine coefficients of like terms:
    $(8 + 3) x^{2} + (1 - 2)x + (3 - 1)$
    $= 11x^{2} - x + 2$
• Subtracting polynomials by distributing the negative sign
  • Example:
  • $(3y^{3} + 2y) - (y^{2} - y + 1)$
  • Compute: $3y^{3} + 2y - y^{2} + y - 1$
  • Collect like terms:
    $3y^{3} - y^{2} + 3y - 1$

Quick terminology and extra notes

• Monomial: one term
• Binomial: two terms
• Trinomial: three terms
• Constant term: a term with degree 0
• Leading term: the term with the highest degree in a polynomial (varies by context)
• Note: Polynomials can involve more than one variable (e.g., $5x^{2}y^{2} + 6x^{2} + 3xy$)