Review and Study Notes: Polynomial and Rational Functions (Chapter 6)

Polynomial and Rational Functions

Factoring Polynomials

Factoring is a fundamental algebraic skill used to simplify expressions, solve equations, and analyze polynomial functions. It involves rewriting a polynomial as a product of simpler polynomials.

• Definition: To factor a polynomial means to express it as a product of its irreducible factors over a given number system (usually real or complex numbers).

• Common Factoring Techniques:

  • Greatest Common Factor (GCF): Factor out the largest common factor from all terms.

  • Grouping: Group terms to factor by pairs.

  • Difference of Squares: $a^2 - b^2 = (a - b)(a + b)$

  • Trinomials: Factor quadratic expressions of the form $ax^2 + bx + c$.

  • Sum/Difference of Cubes: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$, $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

• Example: Factor $2x^2 - 8x$:

  • GCF is $2x$

  • $2x^2 - 8x = 2x(x - 4)$

Solving Polynomial Equations

Solving polynomial equations involves finding all values of the variable that make the equation true. Factoring is often used to set each factor equal to zero and solve for the variable.

• Zero Product Property: If $ab = 0$, then $a = 0$ or $b = 0$.

• Steps:

  1. Set the equation equal to zero.

  2. Factor the polynomial completely.

  3. Set each factor equal to zero and solve.

• Example: Solve $x^2 - 5x = 0$:

  • Factor: $x(x - 5) = 0$

  • Solutions: $x = 0$ or $x = 5$

Rational Expressions and Operations

Rational expressions are quotients of polynomials. Operations include addition, subtraction, multiplication, and division, often requiring factoring and finding common denominators.

• Definition: A rational expression is an expression of the form $\frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials and $Q(x) \neq 0$.

• Key Operations:

  • Multiplication/Division: Multiply/divide numerators and denominators, then simplify.

  • Addition/Subtraction: Find a common denominator, combine numerators, and simplify.

• Example: Simplify $\frac{2x}{x^2 - 4} + \frac{3}{x^2 - 4}$:

  • Common denominator is $x^2 - 4$

  • Combine: $\frac{2x + 3}{x^2 - 4}$

Solving Rational Equations

Rational equations contain rational expressions. Solutions are found by clearing denominators and solving the resulting polynomial equation, checking for extraneous solutions.

• Steps:

  1. Find a common denominator and multiply both sides to clear fractions.

  2. Solve the resulting equation.

  3. Check for extraneous solutions (values that make any denominator zero).

• Example: Solve $\frac{2}{x} = 3$:

  • Multiply both sides by $x$: $2 = 3x$

  • $x = \frac{2}{3}$

  • Check: $x \neq 0$

Polynomial Division and Synthetic Division

Dividing polynomials is useful for simplifying expressions and finding roots. Synthetic division is a shortcut for dividing by linear factors.

• Long Division: Divide as with numbers, subtracting multiples of the divisor.

• Synthetic Division: Used for divisors of the form $x - c$.

• Example: Divide $x^3 - 6x^2 + 11x - 6$ by $x - 2$ using synthetic division.

Applications and Problem Solving

Factoring and rational expressions are used in solving real-world problems, such as finding dimensions, rates, and other quantities modeled by polynomials.

• Example: If the area of a rectangle is $x^2 - 5x + 6$, and one side is $x - 2$, find the other side:

  • Divide: $\frac{x^2 - 5x + 6}{x - 2} = x - 3$

Summary Table: Factoring Techniques

Technique	Form	Example
GCF	$ax + ay$	$a(x + y)$
Grouping	$ax + bx + ay + by$	$(a + b)x + (a + b)y = (a + b)(x + y)$
Difference of Squares	$a^2 - b^2$	$(a - b)(a + b)$
Trinomials	$x^2 + bx + c$	$(x + m)(x + n)$, where $m + n = b$, $mn = c$
Sum/Difference of Cubes	$a^3  b^3$	$(a - b)(a^2 + ab + b^2)$

Additional info:

• Some problems in the file involve solving equations, simplifying rational expressions, and performing polynomial division, all of which are core Precalculus skills.

• Section numbers (e.g., 6-2, 6-3) suggest alignment with textbook chapters on polynomials and rational functions.