prime factor

(noun)

A factor of a given integer that is also a prime number.

Related Terms

  • factoring
  • factorization
  • factor
  • prime number
  • rational expression
  • like term

(noun)

A factor that is also a prime number.

Related Terms

  • factoring
  • factorization
  • factor
  • prime number
  • rational expression
  • like term

Examples of prime factor in the following topics:

  • Factors

    • Such prime numbers are called prime factors.
    • Therefore, 2 and 3 are prime factors of 6.
    • However, 6 is not a prime factor.
    • In this case, we must reduce 6 to its prime factors as well.
    • We have now found factors for 12 that are all prime numbers.
  • Rational Algebraic Expressions

    • For each of the denominators, we find all the prime factors—i.e., the prime numbers that multiply to give that number.
    • If you are not familiar with the concept of prime factors, it may take a few minutes to get used to. 2⋅2⋅32\cdot 2 \cdot 32⋅2⋅3 is 121212 broken into its prime factors: that is, it is the list of prime numbers that when multiplied together yield 12.
    • Similarly, the prime factors of 30 are 2, 3, and 5.
    • Similarly, any number whose prime factors include a 2, a 3, and a 5 will be a multiple of 30.
    • Finding the prime factors of the denominators of two fractions enables us to find a common denominator.
  • Introduction to Factoring Polynomials

    • Factoring by grouping divides the terms in a polynomial into groups, which can be factored using the greatest common factor.
    • Factor out the greatest common factor, 4x(x+5)+3y(x+5)4x(x+5) + 3y(x+5)4x(x+5)+3y(x+5).
    • The aim of factoring is to reduce objects to "basic building blocks", such as integers to prime numbers, or polynomials to irreducible polynomials.
    • One way to factor polynomials is factoring by grouping.
    • Both groups share the same factor (x+5)(x+5)(x+5), so the polynomial is factored as:
  • Inverses of Composite Functions

  • Factoring General Quadratics

    • We can factor quadratic equations of the form ax2+bx+cax^2 + bx + cax​2​​+bx+c by first finding the factors of the constant ccc.  
    • This leads to the factored form:
    • First, we factor aaa, which has one pair of factors 3 and 2.
    • Then we factor the constant ccc, which has one pair of factors 2 and 4.
    • Using these factored sets, we assemble the final factored form of the quadratic
  • Common Bases of Logarithms

    • They describe musical intervals, appear in formulas counting prime numbers, inform some models in psychophysics, and can aid in forensic accounting.
    • Natural logarithms are closely linked to counting prime numbers (2,3,5,72, 3, 5, 72,3,5,7 ...), an important topic in number theory.
    • The prime number theorem states that for large enough N, the probability that a random integer not greater than N is prime is very close to 1log(N)\frac {1} {log(N)}​log(N)​​1​​.
  • Factoring Perfect Square Trinomials

    • When a trinomial is a perfect square, it can be factored into two equal binomials.
    • It is important to be able to recognize such trinomials, so that they can the be factored as a perfect square.
    • If you are attempting to to factor a trinomial and realize that it is a perfect square, the factoring becomes much easier to do.
    • Since the middle term is twice 4⋅x4 \cdot x4⋅x, this must be a perfect square trinomial, and we can factor it as:
    • Evaluate whether a quadratic equation is a perfect square and factor it accordingly if it is
  • Finding Factors of Polynomials

    • When factoring, things are pulled apart.
    • There are four basic types of factoring.
    • The common factor is 333.
    • This is the simplest kind of factoring.
    • Therefore it factors as (x+5)(x−5)(x+5)(x-5)(x+5)(x−5).
  • Solving Quadratic Equations by Factoring

    • To factor an expression means to rewrite it so that it is the product of factors.
    • The reverse process is called factoring.
    • Factoring is useful to help solve an equation of the form:
    • Again, imagine you want to factor x2−7x+12x^2-7x+12x​2​​−7x+12.
    • We attempt to factor the quadratic.
  • Finding Zeros of Factored Polynomials

    • The factored form of a polynomial reveals its zeros, which are defined as points where the function touches the xxx-axis.
    • The factored form of a polynomial can reveal where the function crosses the xxx-axis.
    • In general, we know from the remainder theorem that aaa is a zero of f(x)f(x)f(x) if and only if x−ax-ax−a divides f(x).f(x).f(x). Thus if we can factor f(x)f(x)f(x) in polynomials of as small a degree as possible, we know its zeros by looking at all linear terms in the factorization.
    • This is why factorization is so important: to be able to recognize the zeros of a polynomial quickly.
    • Use the factored form of a polynomial to find its zeros
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