Problem 65 - Beginning and Intermediate Algebra

Question

Factor by grouping. $$k t+7 t-5 k-35$$

Step-by-Step Solution

Answer

The given expression is factored by grouping as follows: \(kt + 7t - 5k - 35 = (kt + 7t) + (-5k - 35) = t(k + 7) - 5(k + 7) = (k + 7)(t - 5)\)

Step 1: 1. Group the terms

First, we need to group the terms in a way that will make factoring easier. Look for terms that share common factors. In this case, the given expression is: $$kt + 7t - 5k - 35$$ We can group the terms as follows: $$(kt + 7t) + (-5k - 35)$$

Step 2: 2. Factor out the common factors

Now, we need to factor out the common factors from each group. In the first group, there is a common factor of 't', and in the second group, there is a common factor of '-5'. So we can rewrite the expression as: $$t(k + 7) - 5(k + 7)$$

Step 3: 3. Factor out the common binomial factor

We now have a common binomial factor, \((k + 7)\), in both terms. We can factor out this binomial factor and rewrite the expression: $$(k + 7)(t - 5)$$ Now the expression has been factored by grouping. The final result is: $$(k + 7)(t - 5)$$

Key Concepts

Common Factors

In algebra, finding common factors is a crucial step in simplifying expressions and solving equations. A common factor is a term that is shared by all terms in an expression. Identifying these can make the process of factoring much simpler.
For the expression given in the exercise, we have:

• In the first group \((kt + 7t)\), the common factor is \(t\).
• In the second group \((-5k - 35)\), the common factor is \(-5\).

By factoring out these common elements, we transform the expression into a more manageable form, enabling us to identify further factoring opportunities. Always look for common numerical factors and variable factors that can be pulled out to reveal more about the structure of the expression.

Binomial Factor

A binomial factor comes into play when we have grouped terms that share a common binomial expression. Recognizing a binomial factor helps in further simplifying complex algebraic expressions.
After extracting common factors from each part of our grouped expression, we noticed a shared binomial factor: \((k + 7)\). This commonality allows us to further simplify our expression:

• Factor out \((k + 7)\) which appears in both terms: \(t(k + 7)\) and \(-5(k + 7)\).
• This leads to the factorizable form \((k + 7)(t - 5)\).

Thus, the expression becomes more streamlined, showcasing the usefulness of spotting a binomial factor in simplifying algebraic equations.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that together form a mathematical sentence. Understanding how to manipulate them is key to mastering algebra.
In the given exercise, we start with the expression \(kt + 7t - 5k - 35\). To manage such an expression, one should:

• Identify and group terms that can be paired due to shared factors.
• Use the properties of common factors and binomial factors to reduce the complexity of the expression.
• Transform the expression into a factored form, simplifying it.
  

Mastering the art of factoring, especially by grouping, is fundamental in simplifying algebraic expressions and solving algebra problems efficiently.