Prime Factorization

Definition

Prime factorization is the process of expressing a whole number as a product of its prime factors. It involves breaking down a number into a unique combination of prime numbers that, when multiplied together, result in the original number. This concept is fundamental to understanding various topics in elementary algebra, such as finding the greatest common factor, factoring polynomials, and simplifying rational expressions and square roots.

5 Must Know Facts For Your Next Test

1. Prime factorization is used to find the greatest common factor (GCF) of two or more numbers, which is essential for factoring by grouping.
2. The prime factorization of a polynomial's coefficients can help determine the factors of the polynomial, especially when factoring trinomials of the form $x^2 + bx + c$.
3. Prime factorization is a key step in the general strategy for factoring polynomials, as it helps identify common factors and potential binomial factors.
4. When adding or subtracting rational expressions with unlike denominators, the prime factorization of the denominators is used to find the least common denominator.
5. The prime factorization of a number is used to simplify square roots by identifying perfect squares within the radicand.

Review Questions

• Explain how prime factorization is used to find the greatest common factor (GCF) of two or more numbers.
  • To find the GCF of two or more numbers using prime factorization, you first express each number as a product of its prime factors. The GCF is then the product of the common prime factors, raised to the lowest power they appear in any of the numbers. By identifying the shared prime factors, you can determine the largest number that divides each of the original numbers without a remainder, which is the GCF.
• Describe how the prime factorization of a polynomial's coefficients can help in factoring trinomials of the form $x^2 + bx + c$.
  • When factoring trinomials of the form $x^2 + bx + c$, the prime factorization of the constant term $c$ can provide valuable clues. By finding two factors of $c$ that add up to the coefficient $b$, you can often identify the binomial factors of the trinomial. This approach, known as the 'ac-method,' relies on the prime factorization of $c$ to systematically test possible pairs of factors until the correct ones are found.
• Explain how prime factorization is used in the general strategy for factoring polynomials and in simplifying square roots.
  • In the general strategy for factoring polynomials, prime factorization is used to identify common factors among the terms, which is the first step in the factorization process. By breaking down the coefficients into their prime factors, you can recognize patterns and common elements that can be factored out. Similarly, when simplifying square roots, the prime factorization of the radicand is used to identify perfect squares within the expression, allowing you to rewrite the square root in a simpler form.

Related terms

Prime Number: A prime number is a positive integer greater than 1 that has no positive divisors other than 1 and itself.

Composite Number: A composite number is a positive integer that has at least one positive divisor other than 1 or itself.

Fundamental Theorem of Arithmetic: The Fundamental Theorem of Arithmetic states that every positive integer greater than 1 can be written as a unique product of prime numbers.