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Precalculus - Polynomial and Rational Functions

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    Study Guide Precalculus – Polynomial and Rational Functions 1. Factoring Polynomials Factoring is like undoing multiplication . Instead of multiplying expressions together, you break a polynomial into simpler parts whose product gives you the original expression. Factoring is one of the most important skills in algebra. It helps you: • Solve higher - degree equations • Simplify expressions • Understand how polynomials are built You may already know some factoring, but this chapter reviews the key methods you should be comfortable with. 1. Greatest Common Factor (GCF) The greatest common factor is the largest factor that divides every term in a polynomial evenly. If all terms share one or more common factors, you can factor them out just like you would factor numbers. How to find the GCF: 1. Look at the numbers in each term. 2. Look at the variables and take the smallest power they all share. 3. Multiply those together. Example 1: Factor the expressions (a) Each term is divisible by 5 and x , so the GCF is 5x .

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    Study Guide Factor it out: (b) The largest factor shared by all terms is: Factor it out: 2. Factoring by Grouping Sometimes, the greatest common factor is not just a number or variable , but an entire expression in parentheses. In these cases, you: 1. Group terms together. 2. Factor each group. 3. Look for a common binomial factor. Example 2: Factor the expressions (a) Every term contains ((x - 5)). Factor it out:

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    Study Guide (b) There is no single GCF for all four terms, so group them: Factor each group: Now factor out ((x - 2)): 3. Factoring Quadratic Trinomials One of the most common factoring tasks is rewriting a quadratic trinomial as the product of two binomials. When the leading coefficient is 1 Use this method when the expression looks like: Find two numbers that: • Add up to the middle coefficient • Multiply to the constant term Example 3 (a)

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    Study Guide We need two numbers that: • Add to ( - 4) • Multiply to ( - 12) Those numbers are ( - 6) and (2). So the factorization is: (b) The numbers that add to ( - 10) and multiply to (24) are ( - 6) and ( - 4). So: When the leading coefficient is not 1 Now the expression looks like: You still look for two numbers that: • Add to the middle coefficient • Multiply to (a \ times c) Then rewrite the middle term and factor by grouping. (c) Multiply the leading coefficient and constant:

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