Study Guide: Exponents, Polynomials, and Factoring (Chapters 4 and 5)

Exponents and Scientific Notation

Properties of Exponents

Understanding and applying the rules of exponents is fundamental in algebra. These rules allow for the simplification and manipulation of expressions involving powers.

• Product Rule: When multiplying exponential expressions with the same base, add the exponents. $a^m \cdot a^n = a^{m+n}$

• Quotient Rule: When dividing exponential expressions with the same base, subtract the exponents. $\frac{a^m}{a^n} = a^{m-n}$, where $a \neq 0$

• Power Rule: When raising a power to another power, multiply the exponents. $(a^m)^n = a^{mn}$

• Negative Exponents: A negative exponent indicates the reciprocal of the base raised to the positive exponent. $a^{-n} = \frac{1}{a^n}$, where $a \neq 0$

• Zero Exponent: Any nonzero base raised to the zero power is 1. $a^0 = 1$, where $a \neq 0$

Scientific Notation

Scientific notation is a way to express very large or very small numbers using powers of ten.

• Form: $a \times 10^n$, where $1 \leq |a| < 10$ and $n$ is an integer.

• Converting to Scientific Notation: Move the decimal point so that only one nonzero digit remains to the left of the decimal. Count the number of places moved to determine the exponent.

• Converting from Scientific Notation: Move the decimal point to the right (for positive exponents) or left (for negative exponents) according to the value of $n$.

• Multiplying and Dividing: Multiply or divide the decimal parts, then add or subtract the exponents. $(a \times 10^m) \cdot (b \times 10^n) = (a \cdot b) \times 10^{m+n}$ $\frac{a \times 10^m}{b \times 10^n} = \left(\frac{a}{b}\right) \times 10^{m-n}$

Polynomials

Definitions and Terminology

A polynomial is an algebraic expression consisting of terms in the form $ax^n$, where $a$ is a coefficient and $n$ is a non-negative integer.

• Term: Each part of a polynomial separated by a plus or minus sign.

• Coefficient: The numerical factor of a term.

• Degree of a Term: The exponent of the variable in the term.

• Degree of a Polynomial: The highest degree among its terms.

• Constant Term: A term with no variable (degree 0).

Types of Polynomials

• Monomial: A polynomial with one term (e.g., $5x^3$).

• Binomial: A polynomial with two terms (e.g., $x^2 - 4$).

• Trinomial: A polynomial with three terms (e.g., $x^2 + 5x + 6$).

Operations with Polynomials

• Addition and Subtraction: Combine like terms (terms with the same variable and exponent).

• Multiplication: Use the distributive property or special products (e.g., $(a+b)^2$).

• Multiplying a Monomial by a Polynomial: Distribute the monomial to each term in the polynomial.

• Multiplying Binomials: Use the FOIL method (First, Outer, Inner, Last) or distributive property.

• Multiplying Polynomials: Multiply each term in one polynomial by each term in the other, then combine like terms.

Example: Multiplying Binomials

• $(x+2)(x+3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6$

Factoring Polynomials

Factoring Techniques

Factoring is the process of writing a polynomial as a product of its factors.

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

• Factoring by Grouping: Group terms to factor common binomials or monomials.

• Factoring Trinomials: For $ax^2 + bx + c$, find two numbers that multiply to $ac$ and add to $b$.

• Factoring Special Products: Recognize patterns such as difference of squares, perfect square trinomials, and sum/difference of cubes.

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

  • Perfect square trinomial: $a^2 + 2ab + b^2 = (a+b)^2$

  • Sum/difference of cubes: $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$

• Factoring Four-Term Polynomials: Use grouping to factor by pairs.

Solving Quadratic Equations by Factoring

To solve quadratic equations by factoring, set the equation to zero, factor the quadratic, and use the zero product property.

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

• Example: $x^2 - 5x + 6 = 0$ Factor: $(x-2)(x-3) = 0$ Solutions: $x = 2$ or $x = 3$

Applications: Pythagorean Theorem

Some applied problems involve quadratic equations that can be solved by factoring, such as those using the Pythagorean Theorem.

• Pythagorean Theorem: $a^2 + b^2 = c^2$ (for right triangles)

• Set up the equation, rearrange to standard form, factor, and solve for the unknown.

Summary Table: Key Polynomial Concepts

Concept	Definition	Example
Monomial	One term	$7x^3$
Binomial	Two terms	$x^2 - 4$
Trinomial	Three terms	$x^2 + 5x + 6$
Degree	Highest exponent	Degree of $3x^4 + 2x^2$ is 4
GCF	Greatest common factor	GCF of $6x^2$ and $9x$ is $3x$

Additional info:

• Some content and examples have been expanded for clarity and completeness.