root

Examples of root in the following topics:

• Roots of Complex Numbers

• Radical Functions

  • An expression with roots is called a radical function, there are many kinds of roots, square root and cube root being the most common.
  • If fourth root of 2401 is 7, and the square root of 2401 is 49, then what is the third root of 2401?
  • If a root is defined as the $n$th root of $x$, it is represented as $\sqrt [ n ]{ x } = r$ .
  • Roots do not have to be square.
  • Irrational numbers also appear when attempting to take cube roots or other roots.

• Introduction to Radicals

  • Roots are the inverse operation of exponentiation.
  • For now, it is important simplify to recognize the relationship between roots and exponents: if a root $r$ is defined as the $n \text{th}$ root of $x$, it is represented as
  • Because roots are the inverse of exponents, we can cancel out the root in this equation by raising the answer to the nth power:
  • This is read as "the square root of 36" or "radical 36."
  • For example, $\sqrt[4]{a}$ is called the "fourth root of $a$," and $\sqrt[20]{a}$ is called the "twentieth root of $a$."

• The Rule of Signs

  • Finding the negative roots is similar to finding the positive roots.
  • Therefore it has exactly one positive root.
  • where $n$ is the total number of roots in a polynomial, $p$ is the maximum number of positive roots, and $q$ is the maximum number of negative roots.
  • Now we look for negative roots.
  • There are 2 complex roots.

• Zeros of Polynomial Functions with Real Coefficients

  • These are often called the roots of the function.
  • There are many methods to find the roots of a function.
  • Therefore both $-1$ and $2$ are roots of the function.
  • Even though all polynomials have roots, not all roots are real numbers.
  • Some roots can be complex, but no matter how many of the roots are real or complex, there are always as many roots as there are powers in the function.

• The Fundamental Theorem of Algebra

  • where $n > 0$ and $c_n \not = 0$, has at least one complex root.
  • For example, the polynomial $x^2 + 1$ has $i$ as a root.
  • admits one complex root of multiplicity $4$, namely $x_0 = 0$, one complex root of multiplicity $3$, namely $x_1 = i$, and one complex root of multiplicity $1$, namely $x_2 = - \pi$.
  • where $f_1(x)$ is a non-zero polynomial of degree $n-1.$ So if the multiplicities of the roots of $f_1(x)$ add to $n-1$, the multiplicity of the roots of $f$ add to $n$.
  • Therefore, a polynomial of even degree admits an even number of real roots, and a polynomial of odd degree admits an odd number of real roots (counted with multiplicity).

• The Discriminant

  • A root is the value of the $x$ coordinate where the function crosses the $x$-axis.
  • The number of roots of the function can be determined by the value of $\Delta$.
  • If ${\Delta}$ is equal to zero, the square root in the quadratic formula is zero:
  • This means the square root itself is an imaginary number, so the roots of the quadratic function are distinct and not real.
  • Because Δ is greater than zero, the function has two distinct, real roots.

• Integer Coefficients and the Rational Zeros Theorem

  • In algebra, the Rational Zero Theorem, or Rational Root Theorem, or Rational Root Test, states a constraint on rational solutions (also known as zeros, or roots) of the polynomial equation
  • Since any integer has only a finite number of divisors, the rational root theorem provides us with a finite number of candidates for rational roots.
  • The advantage of this is that once we have found a root, we immediately have found the smaller degree polynomial of which we again wish to find the roots and the rational root theorem will provide us with even fewer candidates for this root.
  • Moreover, once we have established a root, we must use division anyway to check whether it is a multiple root.
  • We can use the Rational Root Test to see whether this root is rational.

• Imaginary Numbers

  • There is no such value such that when squared it results in a negative value; we therefore classify roots of negative numbers as "imaginary."
  • A radical expression represents the root of a given quantity.
  • When the radicand (the value under the radical sign) is negative, the root of that value is said to be an imaginary number.
  • Specifically, the imaginary number, $i$, is defined as the square root of -1: thus, $i=\sqrt{-1}$.
  • We can write the square root of any negative number in terms of $i$.

• Solving Problems with Radicals

  • Roots are written using a radical sign, and a number denoting which root to solve for.
  • When none is given, it is an implied square root.
  • Roots are written using a radical sign.
  • If there is no denotation, it is implied that you are finding the square root.
  • Otherwise, a number will appear denoting which root to solve for.