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

Verified step by step guidance

1. Identify the polynomial to factor: \$10m^{4} + 43m^{2} - 9\(. Notice that the terms involve powers of \)m^{4}\(, \)m^{2}$, and a constant.
2. Make a substitution to simplify the expression. Let \(x = m^{2}\). Then the polynomial becomes \$10x^{2} + 43x - 9$.
3. Factor the quadratic polynomial \$10x^{2} + 43x - 9$ by looking for two numbers that multiply to \(10 \times (-9) = -90\) and add to \(43\).
4. Rewrite the middle term \$43x$ as a sum of two terms using the numbers found, then factor by grouping.
5. After factoring in terms of \(x\), substitute back \(x = m^{2}\) to express the factorization in terms of \(m\).

Key Concepts

Polynomial Factoring

Factoring polynomials involves rewriting a polynomial as a product of simpler polynomials or factors. This process helps simplify expressions and solve polynomial equations. Common methods include factoring out the greatest common factor, grouping, and using special formulas like difference of squares or trinomials.

Quadratic Form in Polynomials

Some polynomials can be treated as quadratics by recognizing a substitution, such as letting a variable squared equal a new variable. For example, in 10m^4 + 43m^2 - 9, letting x = m^2 transforms it into a quadratic 10x^2 + 43x - 9, which can be factored using quadratic methods.

Factoring Trinomials

Factoring trinomials involves finding two binomials whose product equals the original trinomial. For quadratics in the form ax^2 + bx + c, this often requires finding two numbers that multiply to ac and add to b. This method is essential for breaking down polynomials into simpler factors.