Problem 79 - Algebra and Trigonometry

Question

Factor the expression completely. $$ y^{4}(y+2)^{3}+y^{5}(y+2)^{4} $$

Step-by-Step Solution

Answer

The factored form is \( y^{4}(y+2)^{3} (y^2 + 2y + 1) \).

Step 1: Identify Common Factors

First, we need to find the common factors shared by both terms in the expression.The first term is \( y^{4}(y+2)^{3} \) and the second term is \( y^{5}(y+2)^{4} \).The common factors are \( y^{4} \) and \( (y+2)^{3} \).We will factor out the greatest common factor, \( y^{4}(y+2)^{3} \), from both terms.

Step 2: Factor Out the Greatest Common Factor

Factor out \( y^{4}(y+2)^{3} \) from both terms in the expression.This yields:\[ y^{4}(y+2)^{3} \left[1 + y(y+2)\right] \].

Step 3: Simplify the Expression Inside the Brackets

Now, simplify the expression within the brackets: \( 1 + y(y+2) \).Expand \( y(y+2) \):\[ y \cdot y + y \cdot 2 = y^2 + 2y \].Thus, the expression becomes:\[ y^{4}(y+2)^{3} (1 + y^2 + 2y) \].

Step 4: Combine and Write the Final Factored Expression

Since the expression inside the brackets cannot be factored further, the completely factored form of the original expression is:\[ y^{4}(y+2)^{3} (y^2 + 2y + 1) \].

Key Concepts

Greatest Common Factor (GCF)

The concept of the greatest common factor, or GCF, is essential when simplifying expressions and especially when factoring polynomials. The GCF of two or more terms is the largest factor that each term shares. Finding the GCF is a crucial step in the factoring process because it allows you to "pull out" the largest expression common to all terms, simplifying the problem.

• To find the GCF of polynomial terms, look at the variables and their exponents. Choose the smallest exponent for each variable that appears in every term.
• Apply this to both the numerical coefficients (although in this exercise, the coefficients are 1) and the variables themselves.

In our exercise, we find the GCF of the terms:

• First term: \(y^{4}(y+2)^{3}\)
• Second term: \(y^{5}(y+2)^{4}\)
• Common factor: \(y^{4}(y+2)^{3}\)

The GCF here is \(y^{4}(y+2)^{3}\), thus simplifying the expression by factoring this out from both terms as the initial step.

Algebraic Expressions

Algebraic expressions consist of variables, numbers, and operations such as addition, subtraction, multiplication, and division. In the process of factoring, we aim to rewrite a complex expression into products of simpler expressions without changing its value.Within our exercise, the expression is composed of polynomial terms, where the polynomial is defined as a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients.

• The expression \(y^{4}(y+2)^{3} + y^{5}(y+2)^{4}\) shows two polynomial terms that need to be factored.
• By identifying and extracting the greatest common factor, we simplify the algebraic process.
• After factoring out \(y^{4}(y+2)^{3}\), the remaining expression is \(1 + y(y+2)\), a simpler algebraic expression than we started with.

Simplification through factoring empowers us to see the overall structure of the expression, making further manipulations more straightforward.

Distributive Property

The distributive property in algebra is an essential tool for expanding and factoring expressions. It allows you to multiply a single term by each term in a parenthesis. Alternatively, it is also used for "un-distributing" or factoring common elements in expressions.This property states that for any numbers or expressions \(a\), \(b\), and \(c\):\[a(b + c) = ab + ac\]In the context of our exercise, after factoring out the GCF from the expression:

• The distributive property is applied when expanding \(y(y+2)\), i.e., \(y \cdot y + y \cdot 2 = y^2 + 2y\).
• This shows how we can break down a distributed product into simpler terms.

Using distribution helps simplify terms inside the brackets, as seen with the steps converting \(1 + y(y+2)\) into \(1 + y^2 + 2y\), making the expression ready to be written in a more compact factored form.