Welcome back, class! In our previous lesson, we learned how to graph polynomial functions. Today, we will focus on factoring polynomial equations and using the factored form to solve them. Factoring is a powerful technique that allows us to break down polynomials into simpler expressions and find the solutions to polynomial equations. Let's dive in!
1. What is Factoring?
Factoring involves expressing a polynomial as a product of simpler expressions. It helps us identify the roots or zeros of the polynomial function.
Example:
Consider the polynomial expression \(f(x) = x^2 - 4x + 3\). Factoring it gives us \(f(x) = (x - 1)(x - 3)\).2. Factoring to Solve Equations:
Factoring is particularly useful when solving polynomial equations. By setting each factor equal to zero, we can find the solutions or values that satisfy the equation.
Example:
Let's solve the equation \(g(x) = 2x^2 - 7x - 3 = 0\) by factoring.Step 1: Factor the polynomial expression if possible. In this case, we can rewrite \(g(x)\) as \((2x + 1)(x - 3) = 0\).
Step 2: Set each factor equal to zero and solve for \(x\). We have two equations: \(2x + 1 = 0\) and \(x - 3 = 0\).
3. Finding the Solutions:
By solving the factored equations, we find the solutions, or roots, of the polynomial equation.
Example:
Solving the factored equations from the previous example, we find \(x = \frac{-1}{2}\) and \(x = 3\) as the solutions to the equation \(g(x) = 2x^2 - 7x - 3 = 0\).4. Checking Solutions:
After finding the solutions, it's important to check if they satisfy the original polynomial equation. Substituting the solutions into the equation should yield zero.
Conclusion:
Factoring is a powerful tool for simplifying polynomials and solving polynomial equations. By breaking down polynomials into simpler expressions, we can identify the roots or zeros of the polynomial functions. Factoring provides a systematic and effective approach to finding solutions. In our next lesson, we will continue our exploration of polynomial functions. Stay tuned for more exciting mathematical concepts!