Other Types of Equations

Square root

The graph of the function $f(x)={\sqrt {x}}$, made up of half a parabola with a vertical directrix

In mathematics, a square root of a number x is a number y such that y2 = x; in other words, a number y whose square (the result of multiplying the number by itself, or y ⋅ y) is x. For example, 4 and −4 are square roots of 16, because 4² = (−4)² = 16. Every nonnegative real number x has a unique nonnegative square root, called the principal square root, which is denoted by ${\sqrt {x}}$ where the symbol $\surd$ is called the radical sign or radix. For example, the principal square root of 9 is 3, which is denoted by ${\sqrt {9}}=3$, because 3² = 3 ⋅ 3 = 9 and 3 is nonnegative. The term (or number) whose square root is being considered is known as the radicand. The radicand is the number or expression underneath the radical sign, in this case 9.

Every positive number x has two square roots: ${\sqrt {x}}$ which is positive, and $-{\sqrt {x}}$ which is negative. Together, these two roots are denoted as $\pm {\sqrt {x}}$ (see ± shorthand). Although the principal square root of a positive number is only one of its two square roots, the designation "the square root" is often used to refer to the principal square root. For positive x, the principal square root can also be written in exponent notation, as x^1/2.

Square roots of negative numbers can be discussed within the framework of complex numbers. More generally, square roots can be considered in any context in which a notion of the "square" of a mathematical object is defined. These include function spaces and square matrices, among other mathematical structures.

Cube root

Plot of $y={\sqrt[{3}]{x}}$. The plot is symmetric with respect to origin, as it is an odd function. At x = 0 this graph has a vertical tangent.

In mathematics, a cube root of a number x is a number y such that y³ = x. All nonzero real numbers, have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. For example, the real cube root of 8, denoted ${\sqrt[{3}]{8}}$, is 2, because 2³ = 8, while the other cube roots of 8 are $-1+i{\sqrt {3}}$ and $-1-i{\sqrt {3}}$. The three cube roots of −27i are

$3i,\quad {\frac {3{\sqrt {3}}}{2}}-{\frac {3}{2}}i,\quad {\text{and}}\quad -{\frac {3{\sqrt {3}}}{2}}-{\frac {3}{2}}i.$

In some contexts, particularly when the number whose cube root is to be taken is a real number, one of the cube roots (in this particular case the real one) is referred to as the principal cube root, denoted with the radical sign ${\sqrt[{3}]{~^{~}}}.$ The cube root is the inverse function of the cube function if considering only real numbers, but not if considering also complex numbers: although one has always $\left({\sqrt[{3}]{x}}\right)^{3}=x,$ the cube root of the cube of a number is not always this number. For example, $-1+i{\sqrt {3}}$ is a cube root of 8, (that is, $(-1+i{\sqrt {3}})^{3}=8$), but $-1+i{\sqrt {3}}\neq 2={\sqrt[{3}]{(-1+i{\sqrt {3}})^{3}}}.$

Rational Equations

In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers; they may be taken in any field K. In this case, one speaks of a rational function and a rational fraction over K. The values of the variables may be taken in any field L containing K. Then the domain of the function is the set of the values of the variables for which the denominator is not zero, and the codomain is L.

The set of rational functions over a field K is a field, the field of fractions of the ring of the polynomial functions over K.

A function $f(x)$ is called a rational function if and only if it can be written in the form

$f(x)={\frac {P(x)}{Q(x)}}$

where $P\,$ and $Q\,$ are polynomial functions of $x\,$ and $Q\,$ is not the zero function. The domain of $f\,$ is the set of all values of $x\,$ for which the denominator $Q(x)\,$ is not zero.

However, if $\textstyle P$ and $\textstyle Q$ have a non-constant polynomial greatest common divisor $\textstyle R$, then setting $\textstyle P=P_{1}R$ and $\textstyle Q=Q_{1}R$ produces a rational function

$f_{1}(x)={\frac {P_{1}(x)}{Q_{1}(x)}},$

which may have a larger domain than $f(x)$, and is equal to $f(x)$ on the domain of $f(x).$ It is a common usage to identify $f(x)$ and $f_{1}(x)$, that is to extend "by continuity" the domain of $f(x)$ to that of $f_{1}(x).$ Indeed, one can define a rational fraction as an equivalence class of fractions of polynomials, where two fractions ${\frac {A(x)}{B(x)}}$ and ${\frac {C(x)}{D(x)}}$ are considered equivalent if $A(x)D(x)=B(x)C(x)$. In this case ${\frac {P(x)}{Q(x)}}$ is equivalent to ${\frac {P_{1}(x)}{Q_{1}(x)}}$.

A proper rational function is a rational function in which the degree of $P(x)$ is less than the degree of $Q(x)$ and both are real polynomials, named by analogy to a proper fraction in $\mathbb {Q}$.

Rational function of degree 3, with a graph of degree 3: $y={\frac {x^{3}-2x}{2(x^{2}-5)}}$

Rational function of degree 2, with a graph of degree 3: $y={\frac {x^{2}-3x-2}{x^{2}-4}}$

Degree

There are several non equivalent definitions of the degree of a rational function.

Most commonly, the degree of a rational function is the maximum of the degrees of its constituent polynomials P and Q, when the fraction is reduced to lowest terms. If the degree of f is d, then the equation

$f(z)=w\,$

has d distinct solutions in z except for certain values of w, called critical values, where two or more solutions coincide or where some solution is rejected at infinity (that is, when the degree of the equation decrease after having cleared the denominator).

In the case of complex coefficients, a rational function with degree one is a Möbius transformation.

The degree of the graph of a rational function is not the degree as defined above: it is the maximum of the degree of the numerator and one plus the degree of the denominator.

In some contexts, such as in asymptotic analysis, the degree of a rational function is the difference between the degrees of the numerator and the denominator.

In network synthesis and network analysis, a rational function of degree two (that is, the ratio of two polynomials of degree at most two) is often called a biquadratic function.

Examples

The rational function

$f(x)={\frac {x^{3}-2x}{2(x^{2}-5)}}$

is not defined at

$x^{2}=5\Leftrightarrow x=\pm {\sqrt {5}}.$

It is asymptotic to ${\tfrac {x}{2}}$ as $x\to \infty .$

The rational function

$f(x)={\frac {x^{2}+2}{x^{2}+1}}$

is defined for all real numbers, but not for all complex numbers, since if x were a square root of $-1$ (i.e. the imaginary unit or its negative), then formal evaluation would lead to division by zero:

$f(i)={\frac {i^{2}+2}{i^{2}+1}}={\frac {-1+2}{-1+1}}={\frac {1}{0}},$

which is undefined.

A constant function such as f(x) = π is a rational function since constants are polynomials. The function itself is rational, even though the value of f(x) is irrational for all x.

Every polynomial function $f(x)=P(x)$ is a rational function with $Q(x)=1.$ A function that cannot be written in this form, such as $f(x)=\sin(x),$ is not a rational function. However, the adjective "irrational" is not generally used for functions.

The rational function $f(x)={\tfrac {x}{x}}$ is equal to 1 for all x except 0, where there is a removable singularity. The sum, product, or quotient (excepting division by the zero polynomial) of two rational functions is itself a rational function. However, the process of reduction to standard form may inadvertently result in the removal of such singularities unless care is taken. Using the definition of rational functions as equivalence classes gets around this, since x/x is equivalent to 1/1.

Absolute Value Equations

The graph of the absolute value function for real numbers

In mathematics, the absolute value or modulus of a real number  x, denoted |x|, is the non-negative value of x without regard to its sign. Namely, |x| = x if x is positive, and |x| = −x if x is negative (in which case −x is positive), and Template:Abs = 0. For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3. The absolute value of a number may be thought of as its distance from zero.

Generalizations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces. The absolute value is closely related to the notions of magnitude, distance, and norm in various mathematical and physical contexts.

The real absolute value function is continuous everywhere. It is differentiable everywhere except for x = 0. It is monotonically decreasing on the interval (−∞, 0] and monotonically increasing on the interval [0, +∞). Since a real number and its opposite have the same absolute value, it is an even function, and is hence not invertible. The real absolute value function is a piecewise linear, convex function.

Both the real and complex functions are idempotent.

Polynomial Equations

• 

  Polynomial of degree 0:
  f(x) = 2

• 

  Polynomial of degree 1:
  f(x) = 2x + 1

• 

  Polynomial of degree 2:
  f(x) = x² − x − 2
  = (x + 1)(x − 2)

• 

  Polynomial of degree 3:
  f(x) = x³/4 + 3x²/4 − 3x/2 − 2
  = 1/4 (x + 4)(x + 1)(x − 2)

• 

  Polynomial of degree 4:
  f(x) = 1/14 (x + 4)(x + 1)(x − 1)(x − 3)
  + 0.5

• 

  Polynomial of degree 5:
  f(x) = 1/20 (x + 4)(x + 2)(x + 1)(x − 1)
  (x − 3) + 2

• 

  Polynomial of degree 6:
  f(x) = 1/100 (x⁶ − 2x ⁵ − 26x⁴ + 28x³
  + 145x² − 26x − 80)

• 

  Polynomial of degree 7:
  f(x) = (x − 3)(x − 2)(x − 1)(x)(x + 1)
  (x + 2)(x + 3)

A polynomial function is a function that can be defined by evaluating a polynomial. More precisely, a function f of one argument from a given domain is a polynomial function if there exists a polynomial

$a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{2}x^{2}+a_{1}x+a_{0}$

that evaluates to $f(x)$ for all x in the domain of f (here, n is a non-negative integer and a₀, a₁, a₂, ..., a_n are constant coefficients). Generally, unless otherwise specified, polynomial functions have complex coefficients, arguments, and values. In particular, a polynomial, restricted to have real coefficients, defines a function from the complex numbers to the complex numbers. If the domain of this function is also restricted to the reals, the resulting function is a real function that maps reals to reals.

For example, the function f, defined by

$f(x)=x^{3}-x,$

is a polynomial function of one variable. Polynomial functions of several variables are similarly defined, using polynomials in more than one indeterminate, as in

$f(x,y)=2x^{3}+4x^{2}y+xy^{5}+y^{2}-7.$

According to the definition of polynomial functions, there may be expressions that obviously are not polynomials but nevertheless define polynomial functions. An example is the expression $\left({\sqrt {1-x^{2}}}\right)^{2},$ which takes the same values as the polynomial $1-x^{2}$ on the interval $[-1,1]$, and thus both expressions define the same polynomial function on this interval.

Every polynomial function is continuous, smooth, and entire.

Graphs

A polynomial function in one real variable can be represented by a graph.

• The graph of the zero polynomial f(x) = 0 is the x-axis.
• The graph of a degree 0 polynomial f(x) = a₀, where a₀ ≠ 0, is a horizontal line with y-intercept a₀
• The graph of a degree 1 polynomial (or linear function) f(x) = a₀ + a₁x, where a₁ ≠ 0, is an oblique line with y-intercept a₀ and slope a₁.
• The graph of a degree 2 polynomial f(x) = a₀ + a₁x + a₂x², where a₂ ≠ 0 is a parabola.
• The graph of a degree 3 polynomial f(x) = a₀ + a₁x + a₂x² + a₃x³, where a₃ ≠ 0 is a cubic curve.
• The graph of any polynomial with degree 2 or greater f(x) = a₀ + a₁x + a₂x² + ⋯ + a_nxⁿ, where a_n ≠ 0 and n ≥ 2 is a continuous non-linear curve.

A non-constant polynomial function tends to infinity when the variable increases indefinitely (in absolute value). If the degree is higher than one, the graph does not have any asymptote. It has two parabolic branches with vertical direction (one branch for positive x and one for negative x).

Polynomial graphs are analyzed in calculus using intercepts, slopes, concavity, and end behavior.

Resources

• Polynomial Equations, Lumen Learning
• Radical Equations, Lumen Learning
• Rational Exponent Equations, Lumen Learning
• Absolute Value Equation, Lumen Learning

Licensing

Content obtained and/or adapted from:

• Polynomial, Wikipedia under a CC BY-SA license
• Square root, Wikipedia under a CC BY-SA license
• Cube root, Wikipedia under a CC BY-SA license
• Rational function, Wikipedia under a CC BY-SA license
• Absolute value, Wikipedia under a CC BY-SA license