Problem 73 - Algebra and Trigonometry

Question

31–76 ? Factor the expression completely. $$ y^{4}(y+2)^{3}+y^{5}(y+2)^{4} $$

Step-by-Step Solution

Answer

The expression factors to \(y^{4}(y+2)^{3}(y+1)^{2}\).

Step 1: Identify Common Factors

Examine both terms of the expression: \(y^{4}(y+2)^{3}\) and \(y^{5}(y+2)^{4}\). Identify the greatest common factor (GCF), which is the common factor of both terms. Here, the common factor is \(y^{4}(y+2)^{3}\).

Step 2: Factor Out the GCF

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

Step 3: Simplify Inside the Parentheses

Simplify the expression inside the parentheses, \(1 + y(y+2)\). First, perform the multiplication: \(y(y+2) = y^2 + 2y\). Then, add \(1\) to get:\[1 + y^2 + 2y = y^2 + 2y + 1\]

Step 4: Recognize and Factor Further

Recognize that \(y^2 + 2y + 1\) is a perfect square trinomial. It can be factored further into:\[(y+1)^2\]

Step 5: Final Factored Form

Substitute \((y+1)^2\) back into the expression where we simplified:\[y^{4}(y+2)^{3}(y+1)^{2}\]This is the completely factored form of the given expression.

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is a crucial concept in factoring polynomials because it helps simplify expressions by identifying common elements. When factoring, you look for the largest expression that is a factor in each term of the polynomial. This step is essential for reducing the polynomial to a simpler form.
In our exercise, we analyzed the expression \(y^{4}(y+2)^{3}+y^{5}(y+2)^{4}\). By examining these terms, we found that \(y^{4}(y+2)^{3}\) appears in both parts of the expression. This shared expression is the GCF here.
Taking out the GCF from both terms is like dividing each term by that factor. In doing so, you pull out the common part and simplify the remaining polynomial. In this exercise, factoring out \(y^{4}(y+2)^{3}\) transforms the expression to \(y^{4}(y+2)^{3}(1 + y(y+2))\), setting the stage for further simplification.

Trinomial Factoring

Trinomial factoring involves breaking down a polynomial of the form \(ax^2 + bx + c\) into the product of two binomials. The process requires recognizing patterns or applying methods such as grouping or using the quadratic formula.
In our specific example, once we factor out the greatest common factor, we are left with the expression inside the parentheses: \(1 + y(y+2)\). Performing the multiplication yields \(y^2 + 2y\), and combining this with the initial 1, we get the trinomial \(y^2 + 2y + 1\).
Factoring trinomials is about finding numbers that multiply to get the last term (constant term) while adding up to the middle term (if they are in the form \(x^2 + bx + c\)). For this trinomial, it fits perfectly into the next core concept: perfect square trinomials, which allows further simplification.

Perfect Square Trinomials

Perfect square trinomials occur when a trinomial is the square of a binomial. Recognizing them makes factoring much easier and leads to cleaner solutions.
A trinomial is a perfect square if it fits the pattern \(a^2 + 2ab + b^2\), which can be factored into \((a+b)^2\). In our exercise, the expression \(y^2 + 2y + 1\) can be rearranged to fit this form, \((y)^2 + 2(y)(1) + (1)^2\).
Once we've identified the pattern, we can square the binomial to find the simple factor: \((y+1)^2\).

• This pattern recognition allows us to continuously simplify expressions as we factor more complex polynomials.
• Mastering this technique will make it easier to tackle similar problems efficiently.

Substituting \((y+1)^2\) back into the factored expression gives us the final, completely factored form: \(y^{4}(y+2)^{3}(y+1)^{2}\).