Question

Completely factor the expression.$13x+6+5x^2$

Short answer

$(x+2)(5x+3)$

Step-by-step solution

Arrange in Standard Form

Rearrange the given expression $13x+6+5x^2$ to the standard form, resulting in $5x^2+13x+6$

Factorize the expression

Trying combinations to factorize $5x^2+13x+6$, we can rewrite it as $5x^2+10x+3x+6$. Group the terms to get $5x(x+2)+3(x+2)$. Taking $x+2$ common, we get $(x+2)(5x+3)$ as the factorized form.

Key concepts

Standard Form

In algebra, the **standard form** is a way of writing down expressions and equations. For polynomials, the standard form represents terms by their degree in descending order.

When we say arrange a polynomial in standard form, we start with the highest degree term and end with the constant term.

For instance, the expression given in the problem is $13x+6+5x^2$. Initially, it's not in standard form because the highest degree term $5x^2$ is at the end.

• To convert this into standard form, it is rearranged to $5x^2+13x+6$.
• This ordering helps in processing the expression more systematically, especially for factorization.

Using a consistent order allows us to apply different algebraic methods more reliably.

Grouping Method

The **grouping method** is a powerful algebraic tool for factorization, especially useful for expressions with four or more terms.

It helps break down complex polynomials into simpler parts. This method involves grouping terms in such a way that makes it easier to factor out a common factor.

• Consider our polynomial in standard form: $5x^2+13x+6$. This is broken down into $5x^2+10x+3x+6$ for easier manipulation.
• Next, the terms are grouped: $5x(x+2)+3(x+2)$. Notice each group now has a common factor: $x+2$.
• Finally, factor out the common term, giving the factorized form: $(x+2)(5x+3)$.

This method simplifies the factorization process when direct factoring seems difficult or when the expression does not easily reveal a common factor at first glance.

Quadratic Expression

A **quadratic expression** is a polynomial of degree two, generally in the form $a{x}^2+bx+c$. The expression used in the exercise $5x^2+13x+6$ is a perfect example. Quadratic expressions are fundamental in algebra due to their frequent appearance in a wide range of problems.

Quadratics can describe parabolas, which are U-shaped graphs, and are also used in various real-world applications like physics and engineering.

• The leading coefficient $a$ influences the width and direction of the parabola.
• Identifying a quadratic expression enables us to apply specific factorization techniques such as the grouping method.

Mastering quadratic expressions is key to solving quadratic equations and understanding further algebraic concepts.

Algebra

**Algebra** is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. It is all about finding the unknown or putting real-life variables into equations and then solving them.

Algebra forms the foundational language of mathematics and uses letters to represent numbers that either have yet to be determined or exist as general numbers.

• The expression $13x+6+5x^2$ involves algebraic techniques for simplification and factorization.
• Understanding various algebraic processes is crucial, such as identifying terms, coefficients, and utilizing methods like factorization.
• Through algebra, we learn problem-solving techniques that apply to various fields including statistics, engineering, and economics.

Algebra helps in understanding the relationships between numbers and in modeling real-world situations to solve problems efficiently.