1 University of Phoenix Material Factoring and Radical Expressions The goal of this week is to introduce the algebraic concept of factoring polynomials and simplifying radical expressions. Think of factoring as breaking a very complicated problem into its simple or basic parts. What Is Factoring? The basic idea of factoring is to rewrite an additionsubtraction problem as a multiplication problem. Factors are the numbers that we multiply. For example, 2 and 5 are the factors of 10 because 2 5 = 10. The addition of 6 and 4 can be written into the multiplication of 2 and 5: In other words, we factored the addition 6 + 4 into the multiplication 2 5. The addition of 6 and 4 could have also been written into the multiplication of 1 x 10; however, the goal is always to find prime factors (go to CME > Building Math Confidence > Phoenix Math Topic 62). The factors 1 and 10 are not prime factors of 10 (the answer of adding 6 and 4). Factoring by GCF The factoring process involves applying the distributive property that states. where a, b, and c are terms in an expression. Here is how 6 + 4 was factored into 2 5 using the distributive property: From the demonstration above, the key is to find the greatest common factor (GCF) of each term to apply the distributive property. This same idea can be used to factor polynomials. See the following examples: GCF = 4 GCF = 5 GCF = 3 GCF = GCF = To learn more about how to find the GCF of two or more terms, see the video "Monomial Greatest Common Factor" from Khan Academy (go to CME > Building Math Confidence > Math Videos > Khan Academy).
2 Factoring by Grouping When a polynomial has four terms (or more) and all the terms have no common factors, we can try using a factoring technique called factoring by grouping. This technique consists of grouping the terms of the polynomial and then factoring each group by its GCF. Here is an example: Group (adding parenthesis) the first two terms and last two terms. Factor by GCF each group. The GCF of the first group is and for the second group is 5. Now the GCF is ( ), so the final factoring is Factoring Trinomials by Grouping In Cognitive Tutor, you will learn how to factor trinomials of the forms and using the Guess and Check Method or Undoing FOIL Method. However, these methods can sometimes be time consuming. An alternate method is to factor trinomials by grouping. The basic idea is to convert a trinomial into a polynomial of four terms, so factoring by grouping can be applied. Here is an example: First, multiply the coefficient of the first term by the last term:. Next, find factors of that, when these are added, the result is. In our case, and. Now, rewrite the middle term of the trinomial using the factors of and. We converted the trinomial into a polynomial with four terms, so we can factor by grouping.
3 Factoring Trinomials by ac-method The ac-method is a standard technique for factoring trinomials of the form ax² + bx + c. This method can be used for all non-prime trinomials of that form that can be written as the product of two binomials. Factor 27x² + 6x 5 using the ac-method. First, we label a, b, and c. a = 27 b = 6 c = -5 We ask what two numbers have a product of ac and a sum of b. ac = (27)(-5) = -135 The factors of -1 are as follows (-3)(45) (3)(-45) (-5)(27) (5)(-27) (-15)(9) (15)(-9) Now, let s see which of these has a sum of the value of b. (-3)+(45)= 42 (3)+(-45)= -42 (-5)+(27)= 22 (5)+(-27)= -22 (-15)+(9)= -6 (15)+(-9)= 6 This is the correct one! So, we rewrite our polynomial as follows 27x² 9x + 15x 5 = 9x(3x1) + 5(3x1) = (9x+5)(3x1)
4 Factoring by Special Methods There are strategies that can help you streamline the factoring process of special trinomials and binomials. The key is knowing how to identify these special polynomials. Trinomials Perfect Squares These are trinomials that have the form or and these can be factored as a binomial square with the form or, respectively. In other words, the first and last terms of the trinomial are perfect squares, and the middle term is twice the product of the square roots of the first and last terms. Example: To learn more about factoring trinomial perfect squares, see the video Factoring Perfect Square Trinomials from Khan Academy (go to CME > Building Math Confidence > Math Videos > Khan Academy). Difference of Squares A difference of squares is a binomial in the form. In other words, it is the subtraction of two square terms. This special binomial can be factored as. Example: To learn more about factoring trinomial perfect squares, see the video Factoring Difference of Squares from Khan Academy (go to CME > Building Math Confidence > Math Videos > Khan Academy). Sum and Difference of Cubes Here is the form of sum and difference of cubes and its factoring: Example: To learn more about factoring trinomial perfect squares, see the video Factoring Sum and Difference of Cubes from Khan Academy (go to CME > Building Math Confidence > Math Videos > Khan Academy). Radical Expressions
5 In Cognitive Tutor, you will learn that the root of a number is written as and it means What number raised to power results in? or put another way,. Examples: A. means Example A is read as the square root of and the answer is because 2 = 36. B. means Example B is read as the cube root of 7 and the answer because 3 = 27. C. means Finally, Example C is read as the fifth root of and the answer is because 5 = 32. These are examples of perfect roots because the roots are integers (see Phoenix Math: Topic 21). There will be many instances when the root of a number is not an integer. For example, has no integer root because there is no integer that you can square to get a result of 12. However, these types of roots can be simplified in radical form by using the product rule of radicals that is stated below. The basic idea consists on rewriting the radicand (expression inside of the radical) as a product of two integers, but one of the integers is a perfect root. Here is an example: Understanding how to find nonperfect roots in radical form is necessary to simplify the process of performing basic operations with radical expressions, which you will learn in Cognitive Tutor. Fractional Exponents The root of a number can also be written using fractional exponents instead of using the radical symbols. Here is how: and Examples: Writing radical expressions with fractional exponents provides an alternative method for performing operations with radicals using the exponent rules you learned in the Algebra 1a course, particularly
6 when operations with radicals seems impossible. For example, multiplying two radical expressions with different indexes: