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Chapter 7: Problem 76

Factor by grouping. What should be the first step in factoring \(3 x y+6 x+15 y+30 ?\)

Short Answer

Expert verified
The first step in factoring \(3xy + 6x + 15y + 30\) is to group the terms with common factors as \((3xy+6x)+(15y+30)\). Next, factor out the common factors within each group, which results in \(3x(y+2) + 15(y+2)\). Finally, factor out the common binomial \((y+2)\), obtaining the factorized form \((y+2)(3x+15)\).

Step by step solution

01

Identify the Similar Terms

The first step would involve identifying and grouping together terms that share a common factor. Here, the given expression is already arranged in such a way that every two terms share a common factor. The groups are the first two terms \(3xy+6x\) and the last two terms \(15y+30\).
02

Factoring each Group

The next step is to factor out the common factor in each grouping. For \(3xy+6x\), the common factor is \(3x\), yielding \(3x(y+2)\). The common factor for \(15y+30\) is \(15\), yielding \(15(y+2)\).
03

Factor Common Binomial

After the two groups have been factored, what will be noticed is that there is now a common binomial factor between the groups, which in this case is \(y+2\). This can now be factored out resulting in the factorized form of the expression, which is \((y+2)(3x+15)\). Therefore, the expression \(3xy + 6x + 15y + 30\) can be factorized as \((y+2)(3x+15)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Factor
A common factor is an important concept when dealing with expressions, especially in factoring problems. Simply put, a common factor is a number or variable that divides two or more terms without leaving a remainder. In the context of factoring by grouping, it's the first thing you look for in each set of terms. Here鈥檚 how to spot it:
  • Identify terms within an expression that have shared components.
  • Observe the numbers and variables in each group.
  • Divide the expression into smaller groups where the common factor of each part can be extracted.
Finding a common factor simplifies expressions, making them easier to factor further.
Binomial Factor
A binomial factor comes into play once the common factor has been taken out of the grouped terms. When you factor out common elements from grouped terms, you often end up with an expression in parentheses鈥攖his is the binomial factor. For instance, in the expression \(3xy + 6x + 15y + 30\), after factoring out the common elements in each group, we get \((y+2)\) as a binomial factor. Recognizing binomial factors allows you to further simplify and restructure polynomials.
Polynomial
A polynomial is essentially a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. Understanding polynomials is foundational for learning algebra, particularly intermediate algebra. Polynomials can have many terms鈥攎ore than two terms are common, especially in problems like factoring by grouping.Key characteristics of polynomials:
  • They consist of terms added together, each made up of a coefficient and variable parts.
  • They can be factored to reveal simpler expressions or factors.
The given expression, \(3xy + 6x + 15y + 30\), is a polynomial with four terms that we simplified through factoring by grouping.
Intermediate Algebra
Intermediate algebra serves as the bridge to get from basic algebra to more advanced topics like calculus. It covers skills and understanding necessary for solving real-world problems using algebraic concepts. Factoring by grouping is an essential technique frequently taught in intermediate algebra. This technique breaks down complex polynomial expressions into manageable parts. Here's why factoring by grouping is a vital skill:
  • It reduces expressions to a simpler and often solvable form.
  • Helps in solving polynomial equations and understanding their roots.
  • Useful for simplifying rational expressions and solving equations.
Building a strong foundation in intermediate algebra prepares you for higher-level math courses and practical applications in science and engineering.

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