Precalculus Study Notes: Exponents and Polynomials

Exponents

Definition and Properties of Exponents

Exponents are a shorthand notation for repeated multiplication of the same number or variable. Understanding the rules of exponents is fundamental in algebra and precalculus.

• Exponent Notation: $a^n$ means multiplying a by itself n times.

• Example: $5^3 = 5 \times 5 \times 5 = 125$

Product Rule for Exponents

The product rule allows you to combine exponents when multiplying like bases.

• Rule: $a^m \times a^n = a^{m+n}$

• Example: $x^2 \times x^3 = x^{2+3} = x^5$

Power Rule for Exponents

The power rule is used when raising a power to another power.

• Rule: $(a^m)^n = a^{m \cdot n}$

• Example: $(x^2)^3 = x^{2 \times 3} = x^6$

Quotient Rule for Exponents

The quotient rule is used when dividing like bases with exponents.

• Rule: $\frac{a^m}{a^n} = a^{m-n}$, where $a \neq 0$

• Example: $\frac{x^5}{x^2} = x^{5-2} = x^3$

Zero Exponent

Any nonzero number raised to the zero power is 1.

• Rule: $a^0 = 1$, where $a \neq 0$

• Example: $7^0 = 1$

Negative Exponents

Negative exponents represent reciprocals.

• Rule: $a^{-n} = \frac{1}{a^n}$, where $a \neq 0$

• Example: $x^{-3} = \frac{1}{x^3}$

Fractional Exponents

Fractional exponents represent roots.

• Rule: $a^{1/n} = \sqrt[n]{a}$

• Example: $16^{1/2} = \sqrt{16} = 4$

Polynomials

Definition and Degree

A polynomial is an expression consisting of variables and coefficients, involving only non-negative integer exponents.

• General Form: $a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$

• Degree: The highest exponent of the variable in the polynomial.

• Example: $3x^4 + 2x^2 - 5x + 7$ has degree 4.

Addition and Subtraction of Polynomials

Combine like terms by adding or subtracting their coefficients.

• Example: $(2x^2 + 3x + 1) + (x^2 - 2x + 4) = (2x^2 + x^2) + (3x - 2x) + (1 + 4) = 3x^2 + x + 5$

Multiplication of Polynomials

Multiply each term in the first polynomial by each term in the second, then combine like terms.

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

Multiplying Two Binomials (FOIL Method)

FOIL stands for First, Outer, Inner, Last, referring to the terms multiplied together.

• Example: $(a + b)(c + d) = ac + ad + bc + bd$

Factoring Polynomials

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.

• Example: $6x^2 + 9x = 3x(2x + 3)$

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

• Example: $ax + ay + bx + by = (a + b)x + (a + b)y = (a + b)(x + y)$

Special Factoring Formulas

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

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

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

Practice Problems (Selected Examples)

• Exponent Practice: $x^2 \times x^3 = x^5$

• Negative Exponent: $x^{-2} = \frac{1}{x^2}$

• Polynomial Addition: $(2x^2 + 3x) + (x^2 - x) = 3x^2 + 2x$

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

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

Summary Table: Exponent Rules

Rule	Formula	Example
Product	$a^m \times a^n = a^{m+n}$	$x^2 \times x^3 = x^5$
Quotient	$\frac{a^m}{a^n} = a^{m-n}$	$\frac{x^5}{x^2} = x^3$
Power	$(a^m)^n = a^{mn}$	$(x^2)^3 = x^6$
Zero Exponent	$a^0 = 1$	$7^0 = 1$
Negative Exponent	$a^{-n} = \frac{1}{a^n}$	$x^{-2} = \frac{1}{x^2}$
Fractional Exponent	$a^{1/n} = \sqrt[n]{a}$	$16^{1/2} = 4$

Summary Table: Special Factoring Formulas

Name	Formula
Difference of Squares	$a^2 - b^2 = (a - b)(a + b)$
Sum of Cubes	$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$
Difference of Cubes	$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

Additional info: Some explanations and examples have been expanded for clarity and completeness, based on standard precalculus curriculum.