Problem 62 - A Graphical Approach to College Algebra

Question

Completely factor each polynomial by substitution. $$m^{4}-3 m^{2}-10$$

Step-by-Step Solution

Answer

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

Step 1: Identify the Substitution

Notice that the polynomial is in terms of \(m^4\), \(m^2\), and constants. We can simplify the process by letting \(n = m^2\). Thus, the given polynomial \(m^4 - 3m^2 - 10\) can be rewritten as \(n^2 - 3n - 10\).

Step 2: Factor the Quadratic Expression

Now, factor the quadratic expression \(n^2 - 3n - 10\). We look for two numbers that multiply to \(-10\) and add to \(-3\). These numbers are \(-5\) and \(+2\). Thus, \(n^2 - 3n - 10\) can be factored as \((n - 5)(n + 2)\).

Step 3: Resubstitute the Original Variable

Substitute back \(n = m^2\) into the factored expression. Replace \(n\) with \(m^2\) to obtain \((m^2 - 5)(m^2 + 2)\).

Step 4: Check Your Work

Expand \((m^2 - 5)(m^2 + 2)\) to ensure it matches the original polynomial. Calculating gives \((m^2)(m^2) + (m^2)(2) - (5)(m^2) - (5)(2) = m^4 - 3m^2 - 10\), confirming the factorization is correct.

Key Concepts

Substitution Method

The substitution method is a powerful tool in polynomial factorization. It simplifies complex expressions by temporarily substituting a part of the expression with a new variable.
For this exercise, we identify parts of the original polynomial expression that form a perfect fit for substitution.
By spotting a repeating polynomial term like \(m^2\) in \(m^4 - 3m^2 - 10\), we choose to substitute \(n = m^2\). This temporarily transforms the polynomial into a simpler quadratic form: \(n^2 - 3n - 10\).
The main advantage of this approach is the simplification of the polynomial, making it easier to manipulate and factor. Once we've applied the substitution to facilitate factorization, we reverse it by substituting back the original variable. This method not only streamlines solving but also makes the analysis more straightforward and organized.

Quadratic Expressions

Quadratic expressions, such as \(n^2 - 3n - 10\) in our substituted equation, are crucial in algebra and occur often. A quadratic expression is generally represented in the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants.
Factoring quadratic expressions involves finding two binomials that multiply to give the original quadratic. In this example, we need two numbers that multiply to \(-10\) (the constant) and sum to \(-3\) (the linear coefficient).
These numbers are \(-5\) and \(+2\), leading to the factors \((n - 5)\) and \((n + 2)\). Successfully factoring quadratics simplifies solving equations as it breaks down complex expressions into easier-to-handle parts. It's a foundational skill that leads to mastering higher-level algebra concepts.

Polynomial Expressions

A polynomial expression represents sums and differences of powers of variables, typically in the form \(ax^n + bx^{n-1} + \, \ldots \, + k\).
In our task, the polynomial \(m^4 - 3m^2 - 10\) was expressed in terms of \(m\), showcasing a variety of powers.
Understanding how these polynomial structures interact and how to manipulate and factor them is a key algebraic skill.

• Recognize patterns in powers to simplify expressions, such as \(m^4 = (m^2)^2\), which helped in substituting and eventually factoring the polynomial.
• Mastering factorizations guarantees a deeper comprehension of how polynomial expressions can be broken down into simpler, more manageable parts.
• Review your outcomes by re-expanding the factors to ensure accuracy, mirroring the original polynomial.

This step confirms that the factorization process was executed correctly, reinforcing confidence in dealing with any polynomial challenges you may encounter.