What is the completely factored form of 2×2 32 – In mathematics, factoring is a fundamental operation that involves expressing an algebraic expression as a product of simpler factors. One of the most common types of factoring is the complete factoring of quadratic expressions, which involves finding the prime factors of the expression.
In this article, we will delve into the concept of complete factoring and demonstrate the steps involved in finding the completely factored form of the expression 2x^2 + 32.
By understanding the principles of complete factoring, we gain valuable insights into the structure and behavior of algebraic expressions, making it an essential skill for students and practitioners of mathematics.
Completely Factored Form of 2x^2 + 32: What Is The Completely Factored Form Of 2×2 32
The completely factored form of an expression is a representation of the expression as a product of its irreducible factors. It is the simplest form of an expression, where no further factorization is possible. In this article, we will discuss the concept of completely factored form and apply it to the expression 2x^2 + 32.
Polynomial Factors
A polynomial is an algebraic expression that consists of a sum of terms, where each term is a constant multiplied by a variable raised to a non-negative integer power. Polynomials can be factored into simpler expressions by identifying common factors.
- Factoring Out Common Factors:If a polynomial has a common factor in all its terms, that factor can be factored out using the distributive property.
- Example:Factor out the common factor 2x from 2x^2 + 4x: 2x(x + 2)
Quadratic Expressions
Quadratic expressions are polynomials of degree 2, which can be written in the standard form ax^2 + bx + c. There are several methods for factoring quadratic expressions:
- Factoring by Grouping:When the quadratic expression has four terms, it can be factored by grouping the first two and last two terms, and then factoring out the greatest common factor from each group.
- Completing the Square:This method involves adding and subtracting a constant to the quadratic expression to create a perfect square trinomial, which can then be factored as a binomial squared.
- Using the Quadratic Formula:This formula can be used to find the roots of a quadratic expression, which can then be used to factor the expression.
Prime Factorization
Prime factorization is the process of expressing a number as a product of its prime factors. Prime numbers are numbers that are divisible only by 1 and themselves.
- Finding Prime Factors:Prime factors can be found by dividing the number by the smallest prime number that divides it, and then repeating the process with the quotient until the quotient is 1.
- Example:Prime factorization of 12: 12 = 2 – 6 = 2 – 2 – 3
Completely Factored Form
The completely factored form of an expression is the product of its irreducible factors. For polynomials, this means factoring out any common factors and then factoring any quadratic expressions or prime numbers.
- Significance:Representing expressions in completely factored form makes it easier to analyze and solve equations and inequalities.
- Example:Completely factored form of x^2 – 4: (x – 2)(x + 2)
Case Study: 2x^2 + 32, What is the completely factored form of 2×2 32
To find the completely factored form of 2x^2 + 32, we can use the following steps:
- Factor out the common factor 2: 2(x^2 + 16)
- Factor the quadratic expression x^2 + 16 using the difference of squares formula: 2(x + 4)(x
4)
Therefore, the completely factored form of 2x^2 + 32 is 2(x + 4)(x- 4) .
Essential FAQs
What is the significance of representing expressions in completely factored form?
Representing expressions in completely factored form is significant because it simplifies calculations, facilitates equation solving, and provides insights into the algebraic structure of the expression.
Can you provide an example of an expression written in completely factored form?
Sure, the expression (x – 2)(x + 3) is written in completely factored form.
What is the difference between factoring and prime factorization?
Factoring involves expressing an expression as a product of factors, while prime factorization involves expressing a number as a product of its prime factors.
