Question

Expressions that occur in calculus are given. Factor each expression completely. $$ 4(x+5)^{3}(x-1)^{2}+(x+5)^{4} \cdot 2(x-1) $$

Short answer

The expression factors to \[ 6(x+5)^3 (x-1) (x+1) \].

Step-by-step solution

Identify Common Factors

Look for common factors in each term of the expression. The common factor in this expression is \( (x+5)^3 (x-1) \).

Factor Out the Common Factor

Factor out the common factor \( (x+5)^3 (x-1) \) from each term in the expression. The expression becomes:\[ 4(x+5)^3 (x-1)^2 + (x+5)^4 \cdot 2(x-1) = (x+5)^3 (x-1) \left[ 4(x-1) + 2(x+5) \right] \]

Simplify Inside the Parentheses

Simplify the expression inside the parentheses: \( 4(x-1) + 2(x+5) \).First distribute: \( 4(x-1) = 4x - 4 \) and \( 2(x+5) = 2x + 10 \).Combine like terms: \[ 4x - 4 + 2x + 10 = 6x + 6 \].

Factor Another Common Term

Factor out the common factor \( 6 \) from \( 6x + 6 \): \[ 6(x + 1) \].

Write the Final Factored Form

Combine all the factors: \[ (x+5)^3 (x-1) (6) (x + 1) \]. The fully factored form of the expression is: \[ 6(x+5)^3 (x-1) (x+1) \].

Key concepts

Common Factor

A common factor is a term that appears in all parts of an expression.
When simplifying algebraic expressions, identifying and factoring out the common factor is essential.
It helps in breaking the expression into simpler components, making further simplification easier.

Polynomial Factoring

Polynomial factoring involves breaking down a polynomial into simpler 'factor' expressions that can be multiplied to produce the original polynomial.
This skill is crucial in solving algebraic equations and simplifying expressions.
Common methods of factoring include grouping, using the distributive property, and recognizing formulations like difference of squares.

Simplifying Expressions

Simplifying expressions is the process of making an expression easier to work with.
This could involve factoring, canceling terms, combining like terms, or using algebraic properties.
In our example, we combined like terms and factored out common factors to reach the simplest form.

Calculus Preparation

Factoring and simplifying expressions is foundational for calculus.
Many calculus problems involve complex expressions that need simplification before applying differentiation or integration.
Mastering these skills enables smoother transition into more advanced calculus topics, such as limits, derivatives, and integrals.