Factor each polynomial. See Example 7. 10m^4+43m^2-9

Verified step by step guidance

1. Identify the polynomial: \(10m^4 + 43m^2 - 9\). Notice that it is a quadratic in form, with \(m^2\) as the variable.
2. Rewrite the polynomial as \(10(m^2)^2 + 43(m^2) - 9\) to highlight its quadratic nature.
3. Use the quadratic formula or factoring techniques to factor the expression. Look for two numbers that multiply to \(10 \times -9 = -90\) and add to \(43\).
4. Once the two numbers are identified, rewrite the middle term \(43m^2\) using these numbers, and factor by grouping.
5. Factor out the greatest common factor from each group and simplify to get the factored form of the polynomial.

Key Concepts

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its simpler components, or factors. This process is essential for solving polynomial equations and simplifying expressions. Common methods include factoring out the greatest common factor, using the difference of squares, and applying the quadratic formula for polynomials of degree two.

Quadratic Form

The given polynomial, 10m^4 + 43m^2 - 9, can be treated 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, ax^2 + bx + c, which can be factored using techniques like the quadratic formula or factoring by grouping.

Greatest Common Factor (GCF)

The greatest common factor is the largest factor that divides all terms of a polynomial. Identifying the GCF is a crucial first step in factoring, as it simplifies the polynomial and makes it easier to factor the remaining terms. In the polynomial 10m^4 + 43m^2 - 9, recognizing any common factors can streamline the factoring process.