Exponents and Polynomials: Properties, Operations, and Applications

Exponents and Their Properties

Definitions and Properties of Exponents

Exponents are a way to represent repeated multiplication of the same number or variable. Understanding their properties is essential for simplifying expressions and solving equations in algebra.

• 1 as an exponent: Any number raised to the power of 1 is itself: $a^1 = a$

• 0 as an exponent: Any nonzero number raised to the power of 0 is 1: $a^0 = 1,\ a \neq 0$

• The Product Rule: When multiplying like bases, add the exponents: $a^m \cdot a^n = a^{m+n}$

• The Quotient Rule: When dividing like bases, subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$

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

• Raising a product to a power: Distribute the exponent to each factor: $(ab)^n = a^n b^n$

• Raising a quotient to a power: Distribute the exponent to both numerator and denominator: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$

0 as an Exponent

Evaluating Expressions with Zero Exponents

Zero exponents simplify expressions and are used frequently in algebraic manipulations.

• Key Point: Any nonzero base raised to the zero power equals 1.

• Example: $-3x^0 = -3 \cdot 1 = -3$

• Example: $(4m)^0 = 1$

The Product Rule

Multiplying Powers with the Same Base

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

• Key Point: Add exponents when multiplying like bases.

• Example: $5a^3 \cdot 3a^4 = (5 \cdot 3)a^{3+4} = 15a^7$

• Example: $(x^2y^6)(6x^3y^3) = 6x^{2+3}y^{6+3} = 6x^5y^9$

The Quotient Rule

Dividing Powers with the Same Base

The quotient rule is used to simplify expressions where like bases are divided.

• Key Point: Subtract exponents when dividing like bases.

• Example: $\frac{5m^6}{m^6} = 5m^{6-6} = 5m^0 = 5$

• Example: $\frac{4p^5q^7}{6p^2q} = \frac{4}{6}p^{5-2}q^{7-1} = \frac{2}{3}p^3q^6$

The Power Rule

Raising a Power to a Power

When an exponent is raised to another exponent, multiply the exponents.

• Key Point: $(a^m)^n = a^{mn}$

• Example: $(x^5)^2 = x^{5 \times 2} = x^{10}$

• Example: $(3x)^2 = 9x^2$

Raising a Product to a Power

Distributing Exponents Over Products

When raising a product to a power, apply the exponent to each factor inside the parentheses.

• Key Point: $(ab)^n = a^n b^n$

• Example: $(a^7b)^2(a^3b^4) = a^{14}b^2a^3b^4 = a^{14+3}b^{2+4} = a^{17}b^6$

• Example: $(x^5y^0)^3 = x^{15}y^0 = x^{15}$

Raising a Quotient to a Power

Distributing Exponents Over Quotients

When raising a quotient to a power, apply the exponent to both the numerator and denominator.

• Key Point: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$

• Example: $\left(\frac{3x^4}{y^3}\right)^3 = \frac{27x^{12}}{y^9}$

• Example: $\left(\frac{a^3b^0}{-4c}\right)^2 = \frac{a^6b^0}{16c^2} = \frac{a^6}{16c^2}$

Negative Exponents and Scientific Notation

Negative Integers as Exponents

Negative exponents represent reciprocals. They are useful for expressing very small numbers and for simplifying algebraic expressions.

• Key Point: $a^{-n} = \frac{1}{a^n}$

• Example: $-4x^{-3} = -4 \cdot \frac{1}{x^3} = -\frac{4}{x^3}$

• Example: $\left(\frac{j}{k}\right)^{-7} = \left(\frac{k}{j}\right)^7 = \frac{k^7}{j^7}$

• Example: $\frac{m^8}{m^{-2}} = m^{8-(-2)} = m^{10}$

• Example: $\frac{(a^2)^3}{b^2a^{-4}} = \frac{a^6}{b^2a^{-4}} = \frac{a^{6-(-4)}}{b^2} = \frac{a^{10}}{b^2}$

Scientific Notation

Scientific notation is a way to express very large or very small numbers using powers of ten. The general form is $N \times 10^m$, where $1 \leq N < 10$ and $m$ is an integer.

• To convert from scientific notation to decimal:

  • If $m$ is positive, move the decimal point right $m$ places.

  • If $m$ is negative, move the decimal point left $|m|$ places.

• Example: $7.893 \times 10^5 = 789300$

• Example: $0.00032 = 3.2 \times 10^{-4}$

• Calculator Example: $(4.26 \times 10^{-6})(8.2 \times 10^4) = 3.5 \times 10^{-1}$ (rounded to two significant digits)

• Calculator Example: $\frac{3.48 \times 10^{-7}}{4.6 \times 10^6} = 7.6 \times 10^{-14}$ (rounded to two significant digits)

Additional info: Positive exponents are used for large numbers, negative exponents for small numbers between 0 and 1.

Polynomials and Polynomial Functions

Terms, Monomials, and Polynomials

A term is a product of constants and/or variables. A monomial is a single term with no division by variables. A polynomial is a sum of monomials.

• Examples of polynomials: $3a^2b^4$, $2l + 2w$

• Not polynomials: $\frac{x+3}{x-40}$, $-3a^{-4}$, $5x^2 + x^{-2}$, $\frac{x^2-1}{2x}$

Degree, Leading Term, and Leading Coefficient

The degree of a term is the sum of the exponents of its variables. The leading term is the term with the highest degree in a polynomial written in descending order. The leading coefficient is the coefficient of the leading term.

• Example: For $4x^2 - 6x^3 - 7 + 8x$, written in descending order: $-6x^3 + 4x^2 + 8x - 7$

• Leading term: $-6x^3$

• Leading coefficient: $-6$

• Degree: $3$

Like Terms

Like terms have exactly the same variables with the same exponents, or are constants.

• Example: $6m + 4m^2 - 10m + 12m^2 + 3 = (4m^2 + 12m^2) + (6m - 10m) + 3 = 16m^2 - 4m + 3$

Evaluating Polynomials

To evaluate a polynomial, substitute the given values for the variables and simplify.

• Example: For $-3x^2 - 4x + 2$ at $x = -2$: $-3(-2)^2 - 4(-2) + 2 = -12 + 8 + 2 = -2$

Addition and Subtraction of Polynomials

Combining Like Terms

To add or subtract polynomials, combine like terms and write the result in descending order of degree.

• Example: $(2x-5) + (4-3x) - (-5x+8) = 2x-5 + 4-3x + 5x-8 = (2x-3x+5x) + (-5+4-8) = 4x - 9$

• Example: $(2-3x+x^2) + (-5+7x-3x^2) = (x^2-3x^2) + (-3x+7x) + (2-5) = -2x^2 + 4x - 3$

• Example: $(7x^2 + 3x - 8) - (x^2 - 4x + 2) = (7x^2 - x^2) + (3x + 4x) + (-8 - 2) = 6x^2 + 7x - 10$

Application: Perimeter of a Rectangle

If a rectangle has length $x$ feet and width $(3x-1)$ feet, the perimeter $P$ is:

• $P = 2(l + w) = 2(x + 3x - 1) = 2(4x - 1) = 8x - 2$

Multiplication of Polynomials

Multiplying Monomials

Multiply the coefficients and add the exponents of like variables.

• Example: $(3a^2)(-a) = 3 \cdot (-1) a^{2+1} = -3a^3$

• Example: $5x^6 \cdot 6x^3 = 30x^{6+3} = 30x^9$

Multiplying a Monomial and a Polynomial

Distribute the monomial to each term in the polynomial.

• Example: $-4m^3(m^2 - 6) = -4m^5 + 24m^3$

• Example: $-(x^2 - 5x - 9) = -x^2 + 5x + 9$

Multiplying Two Polynomials

Distribute each term in the first polynomial to every term in the second polynomial.

• Example: $(4x-3)(x-2) = 4x^2 - 8x - 3x + 6 = 4x^2 - 11x + 6$

• Example: $(6 + 5w)(6 - 5w) = 36 - 30w + 30w - 25w^2 = 36 - 25w^2$

• Example: $(x^2 + 2x - 3)(x + 4) = x^3 + 4x^2 + 2x^2 + 8x - 3x - 12 = x^3 + 6x^2 + 5x - 12$

Application: Area of Squares

If one square has side length $6x$ and another has side length $x$, their total area is:

• Area of first: $A_1 = (6x)^2 = 36x^2$

• Area of second: $A_2 = x^2$

• Total area: $A = 36x^2 + x^2 = 37x^2$

Polynomials in Several Variables

Evaluating Polynomials with Multiple Variables

Substitute the given values for each variable and simplify.

• Example: For $3x + xy^2 + 8x^2$ at $x = -2$, $y = 5$: $3(-2) + (-2)(5)^2 + 8(-2)^2 = -6 - 50 + 32 = -24$

Application: Caloric Needs Formula

The number of calories needed each day by a moderately active man can be estimated by:

• $19.18w + 7h - 9.52a + 92.4$

• For $w = 87$ kg, $h = 185$ cm, $a = 59$ years: $19.18 \times 87 + 7 \times 185 - 9.52 \times 59 + 92.4 = 1668.66 + 1295 - 561.68 + 92.4 = 2494.38$ (rounded to nearest whole calorie)

Like Terms in Several Variables

Like terms must have the same variables with the same exponents, regardless of order.

• Like terms: $3a^3b^2$ and $-\frac{1}{2}b^2a^3$

• Not like terms: $4h^2j^5$ and $7h^5j^2$

Operations with Polynomials in Several Variables

• Subtract: $(5xx + 3x^2 - y) - (x^2 + 2xx - y^2) = (5xx - 2xx) + (3x^2 - x^2) + (-y + y^2) = 3xx + 2x^2 + y^2 - y$

• Multiply: $(2m^2 + 5n)(4m^2 - n) = 8m^4 - 2m^2n + 20m^2n - 5n^2 = 8m^4 + 18m^2n - 5n^2$