Question

Expressions that occur in calculus are given. Factor each expression completely. $$ 2(x+3)(x-2)^{3}+(x+3)^{2} \cdot 3(x-2)^{2} $$

Short answer

\((x+3)(x-2)^{2}(5x + 5)\)

Step-by-step solution

Identify Common Factors

Examine the given expression for common factors in each term. The terms are \(2(x+3)(x-2)^{3}\) and \((x+3)^{2} \cdot 3(x-2)^{2}\). Notice that both terms contain the factors \((x+3)\) and \((x-2)^{2}\).

Factor out Common Factors

Factor out the common terms from both parts of the expression. The greatest common factor (GCF) is \((x+3)(x-2)^{2}\): \[2(x+3)(x-2)^{3} + (x+3)^{2} \cdot 3(x-2)^{2} = (x+3)(x-2)^{2} \left[ 2(x-2) + 3(x+3) \right]\]

Simplify the Remaining Expression

Simplify the expression inside the brackets \[ 2(x-2) + 3(x+3) \] First, distribute the numbers: \[ 2x - 4 + 3x + 9 \] Combine like terms: \[ 5x + 5 \]

Write the Final Factored Form

Now, combine all the factors to write the expression in its completely factored form: \[ (x+3)(x-2)^{2}(5x + 5) \]

Key concepts

greatest common factor

The Greatest Common Factor (GCF) is the largest factor that divides two or more numbers. In algebra, GCF refers to the highest degree of polynomial factors common to each term in an expression.
For example, consider these terms:

• Term 1: \(2(x+3)(x-2)^{3}\)
• Term 2: \((x+3)^{2} \cdot\ 3(x-2)^{2}\)

Both terms have \((x+3)\) and \((x-2)^{2}\) as common factors. Therefore, the GCF is \((x+3)(x-2)^{2}\).
By factoring out the GCF, we can simplify complex expressions and make them easier to work with.

simplifying expressions

Simplifying expressions involves reducing them to their simplest form. This process usually includes eliminating parentheses, combining like terms, and performing arithmetic operations.
In our example, we start with: \[2(x+3)(x-2)^{3} + (x+3)^{2} \cdot\ 3(x-2)^{2}\]Once the GCF \((x+3)(x-2)^{2}\) is factored out, the expression inside the brackets are simplified as follows:

First distribute the numbers: \[2(x-2) + 3(x+3)\] So, it becomes: \[2x - 4 + 3x + 9\] Combining the like terms gives us: \[5x + 5\] Now, the expression is simplified to its final factored form: \((x+3)(x-2)^{2}(5x + 5)\)

distributive property

The Distributive Property states that \[a(b + c) = ab + ac\]. This property allows us to multiply a single term by each term inside a bracket.
In our example, we applied the distributive property while simplifying \[2(x-2) + 3(x+3)\]Distribute 2 to both \(x\) and \(-2\): \[(2 \cdot x - 2 \cdot 2 = 2x - 4)\] Now distribute 3 to both \(x\) and \(3\): \[(3 \cdot x + 3 \cdot 3 = 3x + 9)\] By carefully applying the distributive property, we combined terms to reach \[(5x + 5)\].

combining like terms

Combining like terms means summing terms with the same variable raised to the same power.
Only like terms can be combined. For example, \[2x\] and \[3x\] are like terms, whereas \[2x\] and \[3\] are not. In our simplified expression, after distributing, we had: \[2x - 4 + 3x + 9\] Like terms \2x\ and \3x\ can be combined by summing their coefficients: \[2x + 3x = 5x\]. The constant terms \-4\ and \9\ can also be combined: \ -4 + 9 = 5 \. Hence, combining the terms results in \[5x + 5\]. Combining like terms is a crucial step in simplifying algebraic expressions efficiently.