Factor each polynomial. See Example 7. m^4-3m^2-10

Verified step by step guidance

1. Identify the polynomial: \( m^4 - 3m^2 - 10 \). Notice that it is a quadratic in form, with \( m^2 \) as the variable.
2. Make a substitution to simplify the expression: let \( x = m^2 \). The polynomial becomes \( x^2 - 3x - 10 \).
3. Factor the quadratic \( x^2 - 3x - 10 \) by finding two numbers that multiply to \(-10\) and add to \(-3\).
4. The numbers \(-5\) and \(2\) satisfy these conditions, so rewrite the quadratic as \( (x - 5)(x + 2) \).
5. Substitute back \( m^2 \) for \( x \) to get the factors of the original polynomial: \( (m^2 - 5)(m^2 + 2) \).

Key Concepts

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of simpler polynomials or factors. This process is essential for solving polynomial equations and simplifying expressions. Common methods include factoring out the greatest common factor, using special products, and applying techniques like grouping or the quadratic formula when applicable.

Quadratic Form

The expression m^4 - 3m^2 - 10 can be viewed as a quadratic in terms of m^2. By substituting m^2 with a new variable (e.g., x), the polynomial transforms into a standard quadratic form, making it easier to factor. Recognizing this structure allows for the application of quadratic factoring techniques, such as finding two numbers that multiply to the constant term and add to the linear coefficient.

Zero Product Property

The Zero Product Property states that if the product of two factors equals zero, at least one of the factors must be zero. This principle is crucial when solving polynomial equations after factoring, as it allows us to set each factor equal to zero to find the roots of the polynomial. Understanding this property is fundamental for solving equations and analyzing polynomial behavior.