Problem 28 - Algebra A Combined Function

Question

Factor each trinomial completely. $$ m^{4}+10 m^{2}+25 $$

Step-by-Step Solution

Answer

The trinomial is factored as \((m^2 + 5)^2\).

Step 1: Identify the structure of the trinomial

The given trinomial is \( m^4 + 10m^2 + 25 \). Notice that it follows a quadratic-like structure with respect to \( m^2 \). This suggests that it could be factored as a perfect square trinomial.

Step 2: Rewrite the trinomial in terms of a single variable

To facilitate factoring, let \( x = m^2 \). Therefore, the expression becomes \( x^2 + 10x + 25 \). This is now a standard quadratic trinomial.

Step 3: Recognize the perfect square pattern

Observe that \( x^2 + 10x + 25 \) follows the pattern of a perfect square trinomial, \( a^2 + 2ab + b^2 = (a + b)^2 \). Here, \( a = x \), \( b = 5 \), and \( 2ab = 10x \), so \( (x + 5)^2 \) is the factored form.

Step 4: Substitute back to the original variable

Substitute \( x = m^2 \) back into the factored expression \( (x + 5)^2 \). This gives us \((m^2 + 5)^2\).

Step 5: Verify the factorization by expanding

Expand \((m^2 + 5)^2\) to check if it equals \( m^4 + 10m^2 + 25 \). Expanding gives \( m^4 + 10m^2 + 25 \), confirming that the factorization is correct.

Key Concepts

Trinomial

A trinomial is an algebraic expression composed of three terms. These terms are usually separated by addition or subtraction signs. Trinomials are a specific type of polynomial where the highest exponent of the variable is typically used to determine its degree. In our given expression, \( m^4 + 10m^2 + 25 \), the trinomial has three terms.

• \( m^4 \) is the first term, with the highest degree of 4.
• \( 10m^2 \) is the middle term, containing the variable raised to the power of 2.
• 25 is the constant term.

Recognizing trinomials is an essential skill in algebra, as it is the first step towards more complex operations like factoring.

Perfect Square Trinomial

A perfect square trinomial is a special type of trinomial that can be expressed as the square of a binomial. This happens when the trinomial takes the form \( a^2 + 2ab + b^2 \), where \( a \) and \( b \) are any expressions. Understanding the pattern of a perfect square trinomial is key to factoring it correctly.
In the exercise, the transformed expression \( x^2 + 10x + 25 \) is a perfect square trinomial because it can be rewritten as \( (x + 5)^2 \). Here:

• \( a = x \)
• \( b = 5 \)
• \( 2ab = 10x \), confirming the middle term aligns with the perfect square formula.

Understanding this concept helps simplify complex expressions and solve equations efficiently.

Quadratic Expression

A quadratic expression is any polynomial where the highest power of the variable is two. Typically, it can be expressed in the form \( ax^2 + bx + c \). Quadratic expressions can be factored, which is a crucial skill in solving quadratic equations.
In the solution process, by letting \( x = m^2 \), we transformed the original expression into a more familiar quadratic form, \( x^2 + 10x + 25 \). This manipulation eases the factorization process and aligns it with standard methods for solving quadratic equations. Recognizing the quadratic form is often a pivotal step in addressing complex algebraic problems.

Algebraic Equations

Algebraic equations are mathematical statements that contain one or more variables. The goal is often to find the value(s) of the variable(s) that satisfy the equation. Solving algebraic equations involves a variety of methods, including factorization.
In our specific exercise, the key to solving the algebraic equation involved recognizing and transforming the trinomial into a perfect square. This approach simplifies the expression into a solvable form, like \((m^2 + 5)^2\), which confirms the original trinomial structure when expanded. Initial transformations and understanding of such equations lay the groundwork for finding solutions efficiently.