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Algebra Terms and Definitions

Rules and operations involving powers. Features basic concepts like Base and Power, Laws of Exponents, Exponential Functions (a^x), and applications in growth/decay. Includes special cases like Zero, Negative, and Fractional exponents.
Go to Exponents section →
Functions that determine the exponent needed for a base to reach a number. Includes Natural Logarithm (base e), Common Logarithm (base 10), Binary Logarithm (base 2), and their properties. Covers logarithmic functions, equations, identities and transformations.
Go to Logarithms section →
Go to Polynomials section →
Core concepts and operations with roots. Key terms include Square Root (x where x² = n), Cube Root (x where x³ = n), Radical Symbol (√), Perfect Squares/Cubes, and methods of simplification. Covers both real and imaginary roots, radical expressions, and related operations.
Go to Roots section →

Roots

Square Root



Definition:

For a number n, its square root is x where x² = n. Denoted as n\sqrt{n} or n1/2n^{1/2}

- Two values: positive and negative
- Real only if n ≥ 0
- (√n)² = n for n ≥ 0
- √(a·b) = √a·√b
- √(a/b) = √a/√b for b > 0
Common square roots:
√4 = ±2
√9 = ±3
√2 ≈ 1.4142 (irrational)
√0 = 0
√(-1) = i (imaginary)

Cube Root



Definition:

For a number n, its cube root is x where x³ = n. Denoted as n3\sqrt[3]{n} or n1/3n^{1/3}

- Only one real value
- Exists for all real numbers
- (∛n)³ = n
- ∛(a·b) = ∛a·∛b
- ∛(a³) = a
Common cube roots:
∛8 = 2
∛27 = 3
∛(-8) = -2
∛1 = 1
∛0 = 0

Radical Symbol



Definition:

√ for square root, xn\sqrt[n]{x} for nth root where n is the index and x is the radicand

- Index defaults to 2 if omitted
- Index n means nth root
- Radicand is expression under radical
- Can be nested (compound radicals)
Different radical notations:
√x = square root
∛x = cube root
x4\sqrt[4]{x} = fourth root
xn\sqrt[n]{x} = nth root
Index n must be positive integer ≥ 2

Radicand



Definition:

The expression x under the radical sign in xn\sqrt[n]{x}. The value we're finding the root of

- Can be any real number for odd roots
- Must be non-negative for even roots
- Can be variable expression
- Can contain other radicals
In these expressions, radicand is:
√16: 16
∛(-27): -27
x+2\sqrt{x+2}: x+2
814\sqrt[4]{81}: 81

Index (or Degree)



Definition:

The value n in xn\sqrt[n]{x} indicating which root to take (square, cube, fourth, etc)

- Must be positive integer ≥ 2
- Determines number of roots
- Even index: requires non-negative radicand
- Odd index: allows negative radicand
Common indices:
2 (√): square root
3 (∛): cube root
4: fourth root
n: nth root

Principal Root



Definition:

For an even root, the non-negative root out of all possible values. For odd roots, the real root

- Always unique
- Used by default for radical symbol
- Non-negative for even roots
- Same sign as radicand for odd roots
Principal roots:
√4 = 2 (not -2)
∛(-8) = -2
164\sqrt[4]{16} = 2
x2\sqrt{x^2} = |x|

Perfect Square



Definition:

A number n = k² where k is an integer. Also called square number

- Always non-negative
- Integer square root
- Square ends in 0,1,4,5,6,9
- Distance between consecutive grows by 2
First perfect squares:
0 = 0²
1 = 1²
4 = 2²
9 = 3²
16 = 4²
25 = 5²

Perfect Cube



Definition:

A number n = k³ where k is an integer. Also called cubic number

  • - Integer cube root
    - Alternates between odd/even
    - Growing gaps between consecutive
First perfect cubes:
-8 = (-2)³
-1 = (-1)³
0 = 0³
1 = 1³
8 = 2³
27 = 3³

Nth Root



Definition:

Value x where xn=ax^n = a, denoted as an\sqrt[n]{a} or a1/na^{1/n}

- n must be positive integer
- Even n requires a ≥ 0
- Odd n allows any real a
- Principal root is default
Common nth roots:
164=2\sqrt[4]{16} = 2
325=2\sqrt[5]{32} = 2
646=2\sqrt[6]{64} = 2

Radical Expression



Definition:

Mathematical expression containing one or more radicals xn\sqrt[n]{x}. Can include coefficients, variables, and operations

- Contains at least one radical
- May have coefficients
- Can include variables
- Can be simplified
Radical expressions:
232\sqrt{3}
x+y\sqrt{x} + \sqrt{y}
32x33\sqrt[3]{2x}
82\frac{\sqrt{8}}{\sqrt{2}}

Simplifying Radicals



Definition:

Converting radical to equivalent form with smallest possible radicand and rational coefficients outside

- Factor radicand
- Remove perfect nth powers
- Combine like radicals
- Rationalize denominators
Steps to simplify:
12=4⋅3=43=23\sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4}\sqrt{3} = 2\sqrt{3}
543=27⋅23=323\sqrt[3]{54} = \sqrt[3]{27 \cdot 2} = 3\sqrt[3]{2}

Nested Radicals



Definition:

Expression containing radicals inside other radicals: a+b\sqrt{a + \sqrt{b}}

- Multiple radical layers
- Can often be simplified
- Harder to manipulate
- Special denesting methods
Nested forms:
5+24\sqrt{5 + \sqrt{24}}
3+1+2\sqrt{3 + \sqrt{1 + \sqrt{2}}}
2+53\sqrt[3]{2 + \sqrt{5}}

Surd



Definition:

An irrational root of a rational number. A radical that cannot be simplified to a rational number

- Irrational number
- Cannot be further simplified
- Root of rational number
- Contains no imaginary parts
Common surds:
2\sqrt{2}
3\sqrt{3}
53\sqrt[3]{5}
272\sqrt{7} (mixed surd)

Radical Equation



Definition:

Equation containing variable(s) under radical sign: x=a\sqrt{x} = a or f(x)=g(x)\sqrt{f(x)} = g(x)

- Check for extraneous solutions
- Square both sides carefully
- Domain restrictions apply
- May have multiple steps
Solving x−1=2\sqrt{x-1} = 2:
x−1=2\sqrt{x-1} = 2
(x−1)2=22(\sqrt{x-1})^2 = 2^2
x−1=4x-1 = 4
x=5x = 5

Fractional Exponents



Definition:

Root expressions written as powers: x1n=xnx^{\frac{1}{n}} = \sqrt[n]{x} and xmn=(xn)mx^{\frac{m}{n}} = (\sqrt[n]{x})^m

- Numerator = power
- Denominator = root
- Follow exponent rules
- Equivalent to radicals
x12=xx^{\frac{1}{2}} = \sqrt{x}
x13=x3x^{\frac{1}{3}} = \sqrt[3]{x}
x23=(x3)2x^{\frac{2}{3}} = (\sqrt[3]{x})^2

Rationalizing The Denominator



Definition:

Multiplying numerator and denominator by radical term to eliminate radicals in denominator: ab⋅bb=abb\frac{a}{\sqrt{b}} \cdot \frac{\sqrt{b}}{\sqrt{b}} = \frac{a\sqrt{b}}{b}

- Single radical: multiply by itself
- Binomial with radical: multiply by conjugate
- Higher order roots: use appropriate root
12=22\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}

13+2=3−2(3+2)(3−2)\frac{1}{\sqrt{3} + \sqrt{2}} = \frac{\sqrt{3} - \sqrt{2}}{(\sqrt{3} + \sqrt{2})(\sqrt{3} - \sqrt{2})}

Irrational Root



Definition:

A root that yields an irrational number - cannot be expressed as p/q where p,q are integers, q≠0

- Non-terminating, non-repeating decimal
- Cannot be written as ratio of integers
- Often proved irrational by contradiction
Common irrational roots:
2≈1.4142135...\sqrt{2} ≈ 1.4142135...
3≈1.7320508...\sqrt{3} ≈ 1.7320508...
23≈1.2599210...\sqrt[3]{2} ≈ 1.2599210...

Root Approximation



Definition:

Methods to find approximate values of roots: Newton's method: xn+1=xn−f(xn)f′(xn)x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}

- Newton's method
- Binary search
- Calculator estimation
- Numerical algorithms
2\sqrt{2} approximation:
1.4 → 1.414 → 1.4142 → 1.41421

Newton's method for a\sqrt{a}:
xn+1=12(xn+axn)x_{n+1} = \frac{1}{2}(x_n + \frac{a}{x_n})

Conjugate Pair



Definition:

For expression a + √b, its conjugate is a - √b. Product is a² - b. Used to rationalize denominators

- Product removes radicals
- Sum × difference formula
- Preserves value when multiplying num/denom
Conjugate pairs:
a+ba + \sqrt{b} and a−ba - \sqrt{b}
2+3\sqrt{2} + \sqrt{3} and 2−3\sqrt{2} - \sqrt{3}
(a+b)(a−b)=a2−b(a + \sqrt{b})(a - \sqrt{b}) = a^2 - b

Logarithmic Connection



Definition:

an=eln⁡(a)n\sqrt[n]{a} = e^{\frac{\ln(a)}{n}} and an=b  ⟺  a=bn\sqrt[n]{a} = b \iff a = b^n

- Roots as exponentials
- Natural log connection
- Change of base formula
- Solving using logs
x=eln⁡(x)2\sqrt{x} = e^{\frac{\ln(x)}{2}}
x3=eln⁡(x)3\sqrt[3]{x} = e^{\frac{\ln(x)}{3}}
ln⁡(x)=12ln⁡(x)\ln(\sqrt{x}) = \frac{1}{2}\ln(x)

Higher-Order Roots



Definition:

nth root where n > 3: an\sqrt[n]{a} is value x where x^n = a

- n can be any positive integer
- Even n requires a ≥ 0
- Odd n allows any real a
- Multiple complex roots
Fourth root: 164=2\sqrt[4]{16} = 2
Fifth root: 325=2\sqrt[5]{32} = 2
General: ann=∣a∣\sqrt[n]{a^n} = |a| for even n

Imaginary Root



Definition:

For negative real number -a, its square root is i√a where i = √(-1). Higher even roots also yield imaginary results

- Occur when taking even roots of negatives
- Involve imaginary unit i
- Come in conjugate pairs
- Real when n is odd
−4=2i\sqrt{-4} = 2i
−9=3i\sqrt{-9} = 3i
−164=2(1+i)\sqrt[4]{-16} = \sqrt{2}(1 + i)

Powers of i:
i2=−1i^2 = -1
i3=−ii^3 = -i
i4=1i^4 = 1

Logarithms

Logarithm



Definition:

For positive numbers b≠1b ≠ 1 and x>0x > 0, logb(x)=ylog_b(x) = y means by=xb^y = x. Written as: log⁡b(x)=y  ⟺  by=x\log_b(x) = y \iff b^y = x

  • - Base b>0b > 0 and b≠1b ≠ 1
    - One-to-one function
    - Inverse of exponential
log⁡2(8)=3\log_2(8) = 3 because 23=82^3 = 8
log⁡3(27)=3\log_3(27) = 3 because 33=273^3 = 27
log⁡10(100)=2\log_{10}(100) = 2 because 102=10010^2 = 100

Base



Definition:

The positive number b ≠ 1 in logarithmic expression log⁡b(x)\log_b(x) or exponential expression bxb^x

- Must be positive
- Cannot equal 1
- Common bases: 10, e, 2
- Determines growth rate
Common bases:
log⁡10(x)\log_{10}(x) (common log)
log⁡e(x)\log_e(x) (natural log, ln)
log⁡2(x)\log_2(x) (binary log)

Exponent



Definition:

The power y in exponential form byb^y or the value of logarithm log⁡b(x)\log_b(x) where by=xb^y = x

- Can be any real number
- Results in logarithm value
- Determines output level
- Key in exponential growth
In 23=82^3 = 8:
Exponent is 3
log⁡2(8)=3\log_2(8) = 3

In 10x=100010^x = 1000:
x=log⁡10(1000)=3x = \log_{10}(1000) = 3

Natural Logarithm



Definition:

Logarithm with base e(≈2.71828...)e (≈ 2.71828...), written as ln(x)ln(x) or log⁡e(x)\log_e(x). Inverse of exponential function exe^x

  • - Used in continuous growth
    - Base e is irrational
    - Standard notation ln(x)ln(x)

ln⁡(e)=1\ln(e) = 1
ln⁡(e2)=2\ln(e^2) = 2
ln⁡(1)=0\ln(1) = 0
ln⁡(ex)=x\ln(e^x) = x

Common Logarithm



Definition:

Logarithm with base 10, written as log(x)log(x) or log⁡10(x)\log_{10}(x). Used for decimal representations

  • - Counts decimal digits
    - Often written without base
    - Standard notation log(x)log(x)
log⁡(100)=2\log(100) = 2
log⁡(1000)=3\log(1000) = 3
log⁡(10n)=n\log(10^n) = n
log⁡(0.01)=−2\log(0.01) = -2

Binary Logarithm



Definition:

Logarithm with base 2, written as log⁡2(x)\log_2(x). Used in computer science and information theory

  • - Measures bits needed
    - Common in computing
    - Standard notation log⁡2(x)\log_2(x)
log⁡2(8)=3\log_2(8) = 3
log⁡2(16)=4\log_2(16) = 4
log⁡2(2n)=n\log_2(2^n) = n
log⁡2(1024)=10\log_2(1024) = 10

Antilogarithm



Definition:

The inverse logarithm function: if y = log⁡b(x)\log_b(x), then antilogb(y)=x=byantilog_b(y) = x = b^y

  • - Same as exponential function
    - Returns original number
    - Preserves base
If log⁡(100)=2\log(100) = 2
then antilog(2) = 100

If ln⁡(x)=3\ln(x) = 3
then antiln(3) = e³

For log⁡2(8)=3\log_2(8) = 3:
antilog₂(3) = 2³ = 8

Characteristic



Definition:

The integer part n of logarithm where log⁡10(x)=n+d\log_{10}(x) = n + d and 0≤d<10 ≤ d < 1

  • - Indicates magnitude
    - Can be negative
    - For base 10 equals exponent in scientific notation
For log⁡(234)=2.369\log(234) = 2.369:
Characteristic = 2

For log⁡(0.0234)=−1.631\log(0.0234) = -1.631:
Characteristic = -2

Mantissa



Definition:

The decimal part d of logarithm where log⁡10(x)=n+d\log_{10}(x) = n + d and 0≤d<10 ≤ d < 1

  • - Always positive
    - Independent of decimal point position
    - Used in log tables
For log⁡(234)=2.369\log(234) = 2.369:
Mantissa = 0.369

For log⁡(0.0234)=−1.631\log(0.0234) = -1.631:
Mantissa = 0.369

Logarithmic Function



Definition:

Function f(x)=log⁡b(x)f(x) = \log_b(x) where b>0,b≠1b > 0, b ≠ 1. Inverse of exponential function g(x)=bxg(x) = b^x
  • x>0x > 0
    - Range: all real numbers
    - Strictly increasing if b>1b > 1
    - Strictly decreasing if 0<b<10 < b < 1
Common forms:
f(x)=ln⁡(x)f(x) = \ln(x)
f(x)=log⁡10(x)f(x) = \log_{10}(x)
f(x)=log⁡2(x)f(x) = \log_2(x)
Characteristic curved shape crossing y-axis at (1,0)

Complex Logarithm



Definition:

For complex z=r(cosθ+isinθ)z = r(cos θ + i sin θ), ln⁡(z)=ln⁡(r)+i(θ+2πn)\ln(z) = \ln(r) + i(θ + 2πn) where n is integer

  • - Has infinite branches
    - Principal value when n=0n = 0
    - Defined except at z=0z = 0
ln⁡(−1)=πi+2πni\ln(-1) = πi + 2πni
ln⁡(i)=πi2+2πni\ln(i) = \frac{πi}{2} + 2πni
Principal value:
ln⁡(−1)=πi\ln(-1) = πi

Discrete Logarithm



Definition:

For integers a,b,ma, b, m, find xx where ax≡b(modm)a^x ≡ b \pmod{m}. Written as log⁡a(b)(modm)\log_a(b) \pmod{m}

  • - Computationally difficult
    - May not exist
    - Modular arithmetic based
In mod 7:
23≡1(mod7)2^3 ≡ 1 \pmod{7}
so log⁡2(1)≡3(mod7)\log_2(1) ≡ 3 \pmod{7}

3x≡4(mod7)3^x ≡ 4 \pmod{7}
x=log⁡3(4)(mod7)x = \log_3(4) \pmod{7}

Logarithmic Scale



Definition:

Scale where values are spaced by powers of base b: positions proportional to log⁡b(x)\log_b(x) rather than x

  • - Compresses large ranges
    - Often uses base 10
    - Shows percentage changes
Common scales:
pH scale (base 10)
Richter scale (base 10)
Decibels (base 10)
Musical octaves (base 2)
Scientific notation, sound intensity, earthquake magnitude

Exponential Form



Definition:

Equivalent expression of log⁡b(x)=y\log_b(x) = y as by=xb^y = x, showing inverse relationship between logarithms and exponents

  • - Used for solving equations
    - Connects exp and log
    - Base remains constant
log⁡2(8)=3  ⟺  23=8\log_2(8) = 3 \iff 2^3 = 8
ln⁡(x)=4  ⟺  e4=x\ln(x) = 4 \iff e^4 = x
log⁡10(1000)=3  ⟺  103=1000\log_{10}(1000) = 3 \iff 10^3 = 1000

Logarithmic Identity



Definition:

Fundamental rules for manipulating logarithms with same base b:
Product rule: log⁡b(xy)=log⁡b(x)+log⁡b(y)\log_b(xy) = \log_b(x) + \log_b(y)
Quotient rule: log⁡b(xy)=log⁡b(x)−log⁡b(y)\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)
Power rule: log⁡b(xn)=nlog⁡b(x)\log_b(x^n) = n\log_b(x)
Change of base: log⁡b(x)=log⁡a(x)log⁡a(b)\log_b(x) = \frac{\log_a(x)}{\log_a(b)}
log⁡(30)=log⁡(2⋅15)=log⁡(2)+log⁡(15)\log(30) = \log(2 \cdot 15) = \log(2) + \log(15)
log⁡2(82)=log⁡2(8)−log⁡2(2)=3−1=2\log_2(\frac{8}{2}) = \log_2(8) - \log_2(2) = 3 - 1 = 2
log⁡(x3)=3log⁡(x)\log(x^3) = 3\log(x)

Logarithmic Expression



Definition:

Mathematical expression containing one or more logarithms, may include variables and other operations

  • - May have variables
    - Can be simplified using log rules
    - Domain restrictions apply
2ln⁡(x)+32\ln(x) + 3
log⁡(x2+1)\log(x^2 + 1)
log⁡2(x)+log⁡2(y)\log_2(x) + \log_2(y)
ln⁡(x)ln⁡(2)\frac{\ln(x)}{\ln(2)}

Logarithmic Equation



Definition:

Equation containing logarithmic expressions that must be solved for variable(s)

  • - Use log properties to simplify
    - Convert to exponential form
    - Check for extraneous solutions
Solving ln⁡(x)=2\ln(x) = 2:
ln⁡(x)=2\ln(x) = 2
eln⁡(x)=e2e^{\ln(x)} = e^2
x=e2x = e^2

Solving log⁡2(x+1)=3\log_2(x+1) = 3:
2log⁡2(x+1)=232^{\log_2(x+1)} = 2^3
x+1=8x + 1 = 8
x=7x = 7

Logarithmic Inequality



Definition:

Inequality containing logarithmic expressions to be solved: log⁡b(x)<k\log_b(x) < k or log⁡b(f(x))>log⁡b(g(x))\log_b(f(x)) > \log_b(g(x))

- Consider base when solving
- Domain restrictions crucial
- Direction changes if base < 1
- Convert to exponential form
Solving ln⁡(x)>2\ln(x) > 2:
ln⁡(x)>2\ln(x) > 2
eln⁡(x)>e2e^{\ln(x)} > e^2
x>e2x > e^2

log⁡2(x)<3\log_2(x) < 3:
x<23x < 2^3
x<8x < 8

Asymptote



Definition:

For logarithmic function f(x)=log⁡b(x)f(x) = \log_b(x), vertical asymptote at x=0x = 0

- Vertical: x = 0
- Function never crosses
- Defines domain boundary
- Different bases, same asymptote
For y=ln⁡(x)y = \ln(x):
- Vertical asymptote: x = 0
- As x → 0⁺, y → -∞
- As x → ∞, y grows slowly

Graph Of A Logarithmic Function



Definition:

Plot of y=log⁡b(x)y = \log_b(x) showing characteristic shape with vertical asymptote and continuous growth

- Domain: x > 0
- Vertical asymptote at x = 0
- Passes through (1,0)
- Continuous and increasing for b > 1

  • - (b,1) where b is base
    - (1b\frac{1}{b},-1) where b is base

Base-Change Rule



Definition:

log⁡a(x)=log⁡c(x)log⁡c(a)\log_a(x) = \frac{\log_c(x)}{\log_c(a)} for any base c>0,c≠1c > 0, c ≠ 1

- Valid for any positive base
- Commonly used with base e or 10
- Preserves function value
- Useful for calculations
log⁡2(x)=ln⁡(x)ln⁡(2)\log_2(x) = \frac{\ln(x)}{\ln(2)}
log⁡3(x)=log⁡10(x)log⁡10(3)\log_3(x) = \frac{\log_{10}(x)}{\log_{10}(3)}

Logarithmic Growth



Definition:

Growth pattern where variable increases by additive constant when input is multiplied by constant: f(cx)=f(x)+kf(cx) = f(x) + k

  • - Inverse of exponential
    - Common in natural processes
    - Scale-invariant growth

  • - pH scale
    - Earthquake magnitude
    - Sound intensity (decibels)

Logarithmic Transformation



Definition:

Converting data by taking logarithm: y=log⁡b(x)y = \log_b(x) to linearize relationships or normalize distributions

- Compresses large ranges
- Normalizes skewed data
- Linearizes exponential relationships
- Preserves order

  • - Data visualization
    - Economic scales
    - Sound measurement

Polynomials

Polynomial



Definition:

An expression consisting of variables, coefficients, and non-negative integer exponents combined using arithmetic operations.

Coefficient



Definition:

A numerical factor that multiplies a variable in a polynomial.

Leading Coefficient



Definition:

The coefficient of the term with the highest degree in a polynomial.

Free Coefficient



Definition:

The constant term in a polynomial with no variable attached.

Polynomial Degree



Definition:

The highest exponent of the variable in a polynomial.

Monic Polynomial



Definition:

A polynomial whose leading coefficient is 1.

Zero Polynomial



Definition:

A polynomial where all coefficients are zero.

Zero Function



Definition:

A function that always evaluates to zero for any input.

Undefined Degree



Definition:

The degree of the zero polynomial, which is not defined.

Minus Infinity Degree



Definition:

An informal term sometimes used to describe the degree of the zero polynomial.

Polynomial Equation



Definition:

An equation that sets a polynomial equal to another expression, typically zero.

Polynomial Addition



Definition:

The sum of two or more polynomials by adding corresponding terms.

Polynomial Subtraction



Definition:

The difference of two polynomials by subtracting corresponding terms.

Polynomial Multiplication



Definition:

The product of two polynomials by distributing terms.

Polynomial Division



Definition:

The process of dividing one polynomial by another, yielding a quotient and remainder.

Dividend



Definition:

The polynomial being divided in a division operation.

Divisor



Definition:

The polynomial by which another polynomial is divided.

Quotient



Definition:

The result of polynomial division before considering the remainder.

Remainder



Definition:

The leftover polynomial after division that cannot be further divided by the divisor.

Rational Function



Definition:

A function expressed as the ratio of two polynomials.

Remainder Theorem



Definition:

A theorem stating that the remainder when a polynomial P(x) is divided by (x - a) is P(a).

Root Of A Polynomial



Definition:

A value of the variable that makes the polynomial equal to zero.

Sum Of Coefficients Rule



Definition:

The sum of all coefficients of a polynomial is found by evaluating it at x = 1.

Factoring



Definition:

Expressing a polynomial as a product of simpler polynomials.

Quadratic Formula



Definition:

A formula used to find the roots of a quadratic polynomial.

Complex Roots



Definition:

Non-real solutions of a polynomial equation, often involving imaginary numbers.

Multiplicity Of A Root



Definition:

The number of times a particular root appears in the factorization of a polynomial.

Fundamental Theorem Of Algebra



Definition:

A theorem stating that every non-constant polynomial has at least one complex root.

Existence Theorem



Definition:

A principle ensuring the existence of at least one solution for a given polynomial equation.

Factorization



Definition:

The decomposition of a polynomial into a product of lower-degree polynomials.

Polynomial Derivative



Definition:

The derivative of a polynomial function, obtained by differentiating each term.

Increased Root



Definition:

A root of a polynomial whose multiplicity is greater than one.

Linear Factorization



Definition:

Expressing a polynomial as a product of linear factors corresponding to its roots.

Educated Guess Theorem



Definition:

A strategy for guessing rational roots of polynomials using integer coefficients.

Integer Coefficients



Definition:

A polynomial where all coefficients are whole numbers.

Rational Root Theorem



Definition:

A theorem that provides a possible set of rational roots for a polynomial with integer coefficients.

Factorization Theorem



Definition:

A theorem stating that polynomials can be factored uniquely over specific number systems.

Polynomial Long Division



Definition:

A division algorithm for polynomials similar to numerical long division.

Multiple Root



Definition:

A root of a polynomial that appears more than once.

Finite Field



Definition:

A mathematical field containing a finite number of elements.

Modular Arithmetic



Definition:

A system of arithmetic where numbers wrap around after reaching a fixed modulus.

Polynomial Factorization In Finite Fields



Definition:

The process of breaking a polynomial into irreducible factors within a finite field.

Quadratic Residue



Definition:

A number that is a square modulo a given prime number.

Polynomial Roots In Finite Fields



Definition:

The solutions to a polynomial equation within a finite field.

Fundamental Theorem Of Algebra (Finite Fields)



Definition:

A theorem stating that a polynomial of degree n over a finite field has at most n roots in that field.

Direct Substitution Method



Definition:

A technique for solving polynomials by directly substituting values.

Quadratic Formula In Finite Fields



Definition:

A modified version of the quadratic formula adapted for finite fields.

Multiple Root In Finite Fields



Definition:

A root of a polynomial in a finite field that has higher multiplicity.

Vieta's Formulas



Definition:

A set of equations relating the coefficients of a polynomial to sums and products of its roots.

Sum Of Roots



Definition:

The sum of the roots of a polynomial, given by Vieta’s formulas.

Product Of Roots



Definition:

The product of the roots of a polynomial, also given by Vieta’s formulas.

Coefficient Comparison



Definition:

A method for finding unknown coefficients by equating polynomials.

Root Multiplicity



Definition:

The number of times a particular root appears in the factorization of a polynomial.

Polynomial Expansion



Definition:

The process of expanding a factored polynomial into standard form.

Sigma Notation



Definition:

A compact way to represent summation, often used in polynomial expressions.

Polynomial Factorization Using Roots



Definition:

The process of factoring a polynomial using its known roots.

Quadratic And Higher-Degree Root Relations



Definition:

Relationships between the roots and coefficients of quadratic and higher-degree polynomials.

Exponents

Exponent



Definition:

The power to which a base is raised in a mathematical expression.

Base



Definition:

The number that is raised to the power of the exponent.

Power



Definition:

The result of raising a base to an exponent.

Exponential Function



Definition:

A function of the form f(x) = a^x, where a > 0 and a ≠ 1.

Exponential Growth



Definition:

A process that increases proportionally to its current value, modeled by an exponential function.

Exponential Decay



Definition:

A process that decreases proportionally to its current value, modeled by an exponential function.

Scientific Notation



Definition:

A method of writing numbers using powers of 10.

Negative Exponent



Definition:

Indicates the reciprocal of the base raised to the corresponding positive exponent (a^(-n) = 1/a^n).

Zero Exponent



Definition:

Any nonzero base raised to the power of zero equals 1 (a^0 = 1).

Fractional Exponent



Definition:

Represents a root, where a^(1/n) = √[n]{a}.

Integer Exponent



Definition:

An exponent that is a whole number.

Exponential Equation



Definition:

An equation in which variables appear in exponents.

Exponential Inequality



Definition:

An inequality involving exponential expressions.

Exponential Series



Definition:

A mathematical expansion of e^x as a sum of terms.

Exponentiation



Definition:

The mathematical operation of raising one quantity to the power of another.

Laws Of Exponents



Definition:

Rules governing exponentiation, such as the product, quotient, and power rules.

Exponential Curve



Definition:

The graph of an exponential function, showing rapid increase or decrease.

Natural Exponential Function



Definition:

The exponential function with base e, f(x) = e^x.

Compound Interest Formula



Definition:

A financial formula based on exponential growth, A = P(1 + r/n)^(nt).

Exponential Notation



Definition:

A shorthand way to write repeated multiplication of the same factor.