prime factor

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. $2\cdot 2 \cdot 3$ is $12$ 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, $4x(x+5) + 3y(x+5)$.
  • 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 $(x+5)$, so the polynomial is factored as:

• Inverses of Composite Functions

• Factoring General Quadratics

  • We can factor quadratic equations of the form $ax^2 + bx + c$ by first finding the factors of the constant $c$.  
  • This leads to the factored form:
  • First, we factor $a$, which has one pair of factors 3 and 2.
  • Then we factor the constant $c$, 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 ($2, 3, 5, 7$ ...), 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 $\frac {1} {log(N)}$.

• 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 $4 \cdot x$, 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 $3$.
  • This is the simplest kind of factoring.
  • Therefore it factors as $(x+5)(x-5)$.

• 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 $x^2-7x+12$.
  • 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 $x$-axis.
  • The factored form of a polynomial can reveal where the function crosses the $x$-axis.
  • In general, we know from the remainder theorem that $a$ is a zero of $f(x)$ if and only if $x-a$ divides $f(x).$ Thus if we can factor $f(x)$ 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