Simplification Problems to Prepare for Calculus

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1 Simplification Problems to Prepare for Calculus In calculus, you will encounter some long epressions that will require strong factoring skills. This section is designed to help you develop those skills. First, consider a simple simplifying problem. To simplify 7, we distribute to remove the parentheses, then combine like terms: To simplify 7, we can still remove the parentheses and combine like terms: When simplifying 7, removing the parentheses is cumbersome because of the large eponents. Similarly, when simplifying 7, removing the parentheses is impossible because of the fractional eponents. What do we do when removing parentheses is cumbersome or impossible? In these cases, we simplify by factoring. Here is an eample of what an epression from calculus might look like: 7 7 Simplifying an epression such as the one above by factoring requires that we find the greatest common factor GCF. Before tackling this problem, we ll begin with two simpler eamples. This addition sign is the reason that this epression is NOT in factored form. We need to write the epression using ONLY multiplication, division, and powers of polynomials. Eample # Factor 7 5 Solution The GCF for the two terms is remember to use the lower power of a variable when the variable appears in more than one term. We then write the GCF outside a set of grouping symbols usually parentheses or brackets, and use the rules for eponents to determine what terms belong inside the grouping symbols We can find the necessary power of by subtracting the eponent in the GCF from the original power of : 7. The GCF is written outside the grouping symbols.

2 We can check our factoring by using the distributive property to be sure that we obtain the original epression. Eample # Factor y 5 y Solution The GCF for the two terms is 5 y. We write the GCF outside a set of grouping symbols, and use the rules for eponents to determine what terms belong inside the grouping symbols. Each of the new eponents is found by taking the eponent of the variable as found in the original epression, then subtracting the eponent used for that same variable in the GCF y 5 y 5 y y y 5 y y The GCF is written outside the grouping symbols. Eample # Simplify by factoring 7 7 Solution Notice that is a common base in this epression and that is the smaller power to which this base appears. Also, is a common base and is the smaller power to which it appears. Therefore, the greatest common factor is. The GCF can be factored out, as follows: [ 7 ] The GCF is placed outside the grouping symbols. Because the problem already had parentheses, this time the grouping symbols were written as brackets. The remaining steps involve simplifying the polynomial inside the brackets: [ 7 ] [ 7 ] Simplifying the eponents. Using the distributive property.

3 8 Combining like terms. The epression has now been simplified as much as possible, while retaining its factored form. Recall that factored form involves only multiplication, division, and powers of polynomials. The net several eamples involve negative eponents. There are two common ways to approach these simplifying problems. Each method will be described briefly, followed by two eamples. Method A: Use the usual rule of lowest eponents for the GCF, factor out the GCF, and simplify the epression inside the grouping symbols. As the last step, rewrite any epressions involving negative eponents as quotients. Eample #A Simplify by factoring Solution Notice that there is a common base of in this epression, appearing with eponents of and. Since is the smaller eponent to which appears, the GCF is. The new eponents are found by taking the corresponding eponent of as found in the original epression, and subtracting the eponent of in the GCF. [ ] The GCF. This factor simplifies to 0. [ ] [ ] [ 7] 7 Simplifying inside the brackets. 7 Interpreting the negative eponent as a reciprocal.

4 Eample #5A Simplify by factoring. Solution Before factoring, notice that the first half of the epression can be simplified: The product of ½ and is. The product of and is. So the original epression simplifies to. When we compare the two parts of the epression, the smaller eponent of the common base is, and the smaller eponent of the common base is. The common factor is therefore, or, more simply,. Simplifying. [ ] This factor simplifies to 0. Factoring out the GCF and subtracting eponents. This factor simplifies to 0. [ ] [ ] Simplifying inside the brackets. [ ] Interpreting the negative eponent as a reciprocal.

5 5 Method B: First, write epressions involving negative eponents as quotients. Find a common denominator, then combine and simplify the numerators. Eample #B Simplify. Solution Rewriting the negative eponents using reciprocals. The least common denominator is. The second fraction was multiplied by. Adding and simplifying in the numerator. 7 Eample #5B Simplify. Solution Before factoring, notice that the first half of the epression can be simplified: The product of -½ and is -. The product of and is. So the original epression simplifies to. Simplifying

6 Writing the negative eponents using reciprocals. The least common denominator is /. The second fraction was multiplied by Adding and simplifying the numerators. Factoring the numerator.

7 7 Simplification Eercises Simplify each epression ] [

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