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Module 3 notes -Polynomial A polynomial is an algebraic expression composed of both variables and constants that can be added, subtracted, multiplied, divided, or be raised to whole number exponents. An addition or subtraction sign separates each term in a polynomial. Each term in a polynomial is a monomial. -Monomial Mono means one. A monomial is an algebraic expression/polynomial consisting of one single term. Some examples of monomials are x, 5x3, -7sy, and 5. -Binomials Bi means two. A binomial is an algebraic expression consisting of two terms. You could also say that a binomial is an expression that is a sum or difference of two monomials. Some examples of binomials are (5x2 – 2), (x + 2), (2x + 1). Trinomials Tri means three. A trinomial is an algebraic expression consisting of three terms. You could also say that a trinomial is an expression that is a sum or difference of three monomials. Some examples of trinomials are (5x2 – 2x + 3), (x2 + 2x + 2), (2x2 – x + 1). To multiply binomials Remember three key concepts. -Key Concept #1 Multiplying involves the following: A positive times a positive = a positive. (+12) x (+2) = (+24) A negative times a negative = a positive. (-12) x (-2) = (+24) A positive times a negative = a negative. (+12) x (-2) = (-24) -Key Concept #2 When multiplying two powers with the same base, add the exponents and keep the base the same. The rule states that na x nb = na+b. Example: n3 x n2 = n2+3 = n5 Key Concept #3 When multiplying monomials, combine all the coefficients, and then the variables. Use the rules above. Example: Simplify 5x2 - 9x2. Notice, these are like terms because the power in both terms is "x2". The coefficient of each term includes its sign; in 5x2, the coefficient is +5, while in -9x2, it is -9. This is very important to remember. Combine the coefficients to simplify the expression. Multiplying Binomials -When multiplying two binomials, we use the acronym FOIL to remember the order. (An acronym is a word formed from the initials of the words in a phrase; for example, NHL-stands for National Hockey League.) FOIL stands for First, Outside, Inside, Last. First means that the first term in both binomials must be multiplied together. Outside means that the two outermost terms must be multiplied together. Inside means that the two innermost terms must be multiplied together. Last means that the last term of both binomials must be multiplied together. -After you have performed these multiplications, combine any like terms. Like terms are two or more terms that have the same variables raised to the same exponent. This involves adding or subtracting the numerical coefficients and combining that with the variable. Example: Multiply (x + 2)(x - 8). Step 1: Multiply the first term in each binomial. This step produces x2. Step 2: Outside: Multiply the outermost terms. This step produces -8x. Step 3: Inside: Multiply the innermost terms. This step produces +2x. Step 4: Last: Multiply the last terms of each binomial. This step produces –16. Therefore: (x + 2) (x – 8) = x2 – 8x +2x – 16 = x2 – 6x – 16 Notice that –8x and +2x are like terms; therefore, after combining those two terms, the resulting trinomial was x2 – 6x – 16. Checking Your Answers To check to see if you have simplified algebraic expressions correctly, substitute numbers for the variables in both the original question and the answer. If they have the same value, you have done the question correctly. -For example, check the following question. Let the value of x be 2 (x = 2). In fact, using a number that is 2 or greater is always a good technique. Sometimes 0 and 1 make your solution seem correct even when it is not, because taking 0 or 1 to a positive exponent does not change the value. Example: Because they both have the same values, your multiplication is correct. -Multiplying polynomials Multiplying polynomials is similar to multiplying binomials. One must multiply every term in one polynomial with every term in the other polynomial. This multiplication illustrates the values, distributive property. is correct. Because they both have the same your multiplication Example: (x – 3)(x2 + 6x – 5) Again, every term in one polynomial must be multiplied to every term in the other. Multiplying each step produces the following. (Notice that like terms are stacked so that combining terms becomes easier.) After the terms are combined from Steps 2, 3, 4, and 5, the resulting polynomial is x3 + 3x2 – 23x + 15. Review Factoring is rewriting an expression as a product of two or more expressions: 2x(x + 1) is the factored form of 2x2 + 2x. Greatest Common Factor (GCF) is the largest number or expression that divides into a given expression without a remainder. The greatest common factor of 12 and 8 is 4 because 4 is the largest number that will divide into both 12 and 8 without remainder. The greatest common factor of 2x and 6xy is 2x because 2x is the greatest term that will divide into 2x and 6xy without remainder. Prime factorization represents a number as a product of its prime factors: 96 is 2 x 2 x 2 x 2 x 2 x 3. Powers are repeated multiplication: n3 means n x n x n. Finding the Common Factor of a Polynomial To find the common factor of a poynomial, follow these steps. Step 1: Factor fully each term in the polynomial. Ignore the sign of each term for now. Include them when the polynomial is rewritten in factored form. Factor coefficients into prime factors, and express any powers as repeated multiplication. Example: 14 x2y3 – 56xy2z2 + 28 xy2 14 x2y3 = 2 ● 7 ● x ● x ● y ● y ● y 56 xy2z2 = 2 ● 2 ● 2 ● 7 ● x ● y ● y ● z ● z 28 xy2 = 2 ● 2 ● 7 ● x ● y ● y Step 2: Examine each fully factored term, and circle the items that are common to each term. In the example, the following circles show factors that are common to each term. Step 3: Multiply the circled items together to get the GCF. Step 4: Rewrite the original polynomial 14 x2y3 – 56 xy2z2 + 28 xy2 in factored form. Notice that the arithmetic symbols have been highlighted in red to show that these do not change when the polynomial is in factored form. To factor, follow this process: This is what you get when you use the division rule. When you divide exponents with the same base, you subtract the exponents. Also, remember that anything taken to the zero exponent has a value of 1. The above expression simplifies to the following: Of course, anything multiplied by 1 is itself, and anything to the exponent 1 can be written without actually showing exponent 1. In other words, x1 = x. Therefore, the step above simplifies to This is in factored form because the last operation is multiplication (See below.) The two factors are (14xy2) and (xy – 4z2 + 2). Step 5: Check by multiplying the factors. If the product is equivalent to the original polynomial, then your work is correct. Notice, in the example above, a lot of detail was provided within each step to eliminate confusion. As you practice, you will become more proficient at putting polynomials in factored form, and you will need fewer steps. Factoring Trinimials The next type of trinomial to be factored has the value of "a" not equal to one, even after the GCF has been removed. To factor ax2 + bx + c, where a ≠ 1, a factoring method called grouping - decomposition is used. The steps used in this type of factoring are the following: Step 1: If possible, remove a GCF. Step 2: Determine all the pairs of factors of the product "ac". Add these factors together. The pair of factors that have a sum of "b" are used in step 3. Step 3: Rewrite the x term as the sum of the two terms with these factors as coefficients. This step "decomposes" the middle term into two pieces. Step 4: Remove a common factor from the first two terms, and then remove a common factor from the last two terms. This is the grouping part of the method. Step 5: There will be another common factor in the terms. Remove the common factor and simplify. You have now factored the trinomial. Example: Factor the following: 2x2 + 11x + 12. Step 1: Remove any common factors. Inspect 2x2, 11x and 12. There is no common factor. Step 2: Determine the correct product (ac) and sum (b). Since a = + 2, c = +12, b = +11, the value of ac is 24. Find the two factors that have a product of ac = + 24 and a sum of b = +11. Factors of +24 1 and 24 –1 and –24 2 and 12 –2 and –12 3 and 8 –3 and –8 4 and 6 –4 and –6 Product (a●c) 24 24 24 24 24 24 24 24 Sum (b) 25 –25 14 –14 11 –11 10 –10 Satisfies both conditions? no no no no yes no no no From the chart, the two factors are 3 and 8 because they satisfy both conditions. Step 3: Decompose the middle term. 2x2 + 11x + 12 becomes 2x2 + 8x + 3x + 12 Step 4: Group by taking a common factor out of first two terms (in green) and last two terms (in pink) and then group. 2x2+8x + 3x + 12 GCF for 2x2 + 8x = 2x GCF for 3x + 12 = 3 2x(x + 4) + 3(x + 4) Notice that each term has (x + 4)(2x + 3). This is the factored form of 2x2 + 11x + 12. Always test the answer by multiplying using FOIL. The result should be equal to the original. Because of this was the original polynomial, you know this trinomial has been factored correctly. -Special factors A perfect square trinomial is the result of squaring a binomial. For example, (x + 3)2 produces the trinomial x2 + 6x + 9. Many students make the error of thinking that (x+3)2 = x2 + 9 because they do not realize (or they forget) that (x + 3)2 = (x + 3)(x + 3). When this is multiplied, a middle term occurs. The method of FOIL demonstrates this. To factor a perfect square trinomial where a=1, follow the process shown in the previous sections. Example: Factor x2 –10x + 25 Because this is a simple trinomial where a=1, find two numbers that multiply to give +25 and add to give –10. _____ x _____ = +25 _____ +_____ = –10 The numbers are –5 and –5. –5 x –5 = +25 –5 + –5 = –10 Rewrite this as factors to get (x – 5)(x – 5), which is (x – 5)2. Sometimes, you need to factor a perfect square trinomial where a does not equal 1. In these cases, recognize that it is a perfect square trinomial meets the following criteria: The first and third terms are perfect squares (example 4, 9, 16, 25, 36, 49, etc.) The second term (b) is equal to the square root of the first term a multiplied by the square root of the third term (C) and then all multiplied by 2. As an example: 16x2 + 24x + 9 is a perfect square because 16 and 9 are both perfect squares. The square root of 16 (which is 4) times the square root of 9 (which is 3) and then all multiplied by 2 is 24. In other words, Therefore, to factor a perfect square trinomial, you have two options: Method A: Continue to factor as any trinomial where A is not equal to 1. Method B: Recognize it to be a perfect square trinomial, and take a short cut. Both methods are shown below. You can choose which you prefer. Example: Factor 16x2 + 24x + 9. Method A: Factoring as a Trinomial Start by trying to remove the GCF, in this case, there is no GCF. Then, find two numbers that multiply to give a●c and add to give b. (a = 16, b = 24, c= 9) a = 16 x 9 = 144 Factors of 144 are (1 x 144), (2 x 72), (3 x 48), (4 x 36), (6 x 19), (8 x 18) and (12 x 12). The only factors that add to 24 are (12 x 12). Write the trinomial as a polynomial with four terms in which the middle term is decomposed: 16x2 + 12x +12x + 9 (Group like terms.) 4x(4x + 3) +3(4x + 3) (Take out common factor.) (4x + 3)(4x + 3) (Write as a perfect square.) Therefore, 16x2 + 24x + 9 = (4x +3)2 Method B: Recognizing and Factoring as a Perfect Square The challenge of this method is to recognize the polynomial as a perfect square. Consider the factored forms of some perfect squares that have been factored. Check to see if there is a pattern. x2 + 16x + 64 = (x + 8)2 49x2 + 14x +1 = (7x + 1)2 x2 + 2x +1 = (x + 1)2 4x2 – 36x + 81 = (2x - 9)2 x2 – 10 x + 25 = (x - 5)2 Have you figured out the pattern? To change a perfect square trinomial into factored form... Factoring a Difference of Two Perfect Squares -Also known as the difference of squares, the difference of perfect squares occurs when a number is squared (multiplied by itself) and then subtracted from another squared number. Example: Factor 16x2 - 25 This expression is a difference of squares because 16x2 is a perfect square, 25 is a perfect square, and the minus sign means difference. When factoring a difference of squares, take the square root of each term in the polynomial to find the terms of the binomials. One factor always has a plus sign; the other has a minus sign. To factor 16x2 – 25, follow the following steps: Therefore the factored form of 16x2 – 25 is (4x + 5)(4x – 5).