Factoring Trinomials: Part 1
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1 Factoring Trinomials: Part 1 Factoring Trinomials (a = 1) We will now learn to factor trinomials of the form a + b + c, where a = 1 Because a is the coefficient of the leading term of the trinomial, this means that we are factoring trinomials of the form + b + c where the leading coefficient is 1 When we multiply two binomials together we use the FOIL Method and the result is often a trinomial Consequently, when we factor a trinomial we are applying the FOIL method in reverse by breaking up the trinomial into two binomials that are factors of the trinomial Eample: Multiply the binomials ( + 3)( + 5) to obtain a trinomial Solution: Use the FOIL Method ( + 3)( + 5) Now, to reverse the process by factoring, we must first notice the following: 1 The c term (15) is the product of the constants in each binomial (35) The b term (8) is the sum of the constants in each binomial (3+5) This leads to the following procedure for factoring a trinomial where a = 1 1 Find all sets of factors of the c term Select the set of factors (with appropriate signs) that will combine to give the b term 3 Place these factors in the parentheses ( + _)( + _) Eample: Factor the trinomial Solution: Since the a value is 1, we need to find factors of 15 that combine to The factors -3 and -5 will work Therefore, we can factor the trinomial as: ( + 3)( + 5) ( 3)( 5)
2 Eample: Factor the trinomial Solution: Since the a value is 1, we need to find factors of -4 that combine to The factors -3 and 8 will work Therefore, we can factor the trinomial as: ( 3)( + 8) Eample: Factor the trinomial Solution: Since the a value is 1, we need to find factors of 18 that combine to The factors and 9 will work Therefore, we can factor the trinomial as: ( + )( + 9) Descending Order Sometimes the trinomial is not written in descending order It is usually easier to factor a trinomial when it is in descending order, therefore first rewrite the trinomial so that it is in descending order Eample: Factor the trinomial Solution: Begin by rearranging the terms so that the polynomial is in descending order Since the a value is 1, we need to find factors of 3 that combine to The factors -4 and -8 will work Therefore, we can factor the trinomial as: ( 4)( 8)
3 Eample: Factor the trinomial y 4 + y Solution: Begin by rearranging the terms so that the polynomial is in descending order y + y 4 Since the a value is 1, we need to find factors of -4 that combine to The factors -4 and 6 will work Therefore, we can factor the trinomial as: y + y 4 ( y 4)( y + 6) Multiple Variables Sometimes a trinomial will have two variables In this case, we factor eactly the same ecept we multiply the nd variable to the constant in each binomial Eample: Factor the trinomial + 4y + 3y Solution: Since there are two variables, the 1 st term in each binomial will contain an and the nd term will contain a y Since the a value is 1, we need to find factors of 3 that combine to The factors 1 and 3 are the only factors Therefore, we can factor the trinomial as: + 4y + 3y ( + 1( y))( + 3( y)) ( + y)( + 3y) Eample: Factor the trinomial 5y 14y Solution: Since there are two variables, the 1 st term in each binomial will contain an and the nd term will contain a y Since the a value is 1, we need to find factors of -14 that combine to
4 The factors and -7 are the only factors Therefore, we can factor the trinomial as: 5y 14y ( + y)( 7y) Greatest Common Factor If the terms of the trinomial have common factors, first factor out the GCF The GCF must be written as part of the final factorization Eample: Factor the trinomial Solution: Since each term has a GCF of, factor out the first ( ) Now we need to factor the remaining trinomial Since the a value is 1, we need to find factors of -18 that combine to The factors -3 and 6 will work Therefore, therefore, the complete factorization is: ( ) ( 3)( + 6) 3 Eample: Factor the trinomial Solution: Since each term has a GCF of 8, factor out the 8 first ( 3 + ) Now we need to factor the remaining trinomial 3 + Since the a value is 1, we need to find factors of that combine to -3 1 The factors - and -1 will work Therefore, therefore, the complete factorization is: 8( 3 + ) 8( )( 1)
5 Leading Term Negative If the leading term is negative, first factor out a negative one (-1) from each term so that the resulting leading term is positive Eample: Factor the trinomial Solution: Since the 1 st term is negative, begin by factoring out a (-1) from each term ( ) Since the a value is 1, we need to find factors of 14 that combine to The factors - and -7 will work Therefore, we can factor the trinomial as: ( ) ( )( 7) Eample: Factor the trinomial Solution: Since the 1 st term is negative, begin by factoring out a (-1) from each term ( ) Since the a value is 1, we need to find factors of -11 that combine to The factors -1 and 11 will work Therefore, we can factor the trinomial as: ( ) ( 1)( + 11)
6 Applications: Eample: Scientists who study genetics use the equation p + pq + q = 1, where p represents a certain dominant gene and q represents a recessive gene Rewrite the equation so that the left side is factored Solution: p + pq + q = 1 ( p + q)( p + q) = 1 Eample: A child throws a stone with an initial velocity of 48 fps from a height of 160 feet above the ground The equation s = 16t 48t models this situation Rewrite the equation so that the right side is factored completely Solution: s = 16t s = 16( t 48t t 10) s = 16( t + 5)( t ) Eample: According to specifications, a bo manufacturer makes a closed bo with length that is 4 inches longer than the height and width that is 3 inches longer than the height If the volume of this bo can be represented by the equation V = rewrite the equation in terms of its length, width, and height Solution: We must factor the equation and then write it as V=LWH Let represent the height of the bo 3 V = V = ( ) V = ( + 3)( + 4) Therefore, the height, the length is (+4), and the width is (+3) Eample: A statistician found that the cost in dollars for a company to produce units of a certain product can be approimated byc = Factor the right side of this equation Solution: C = C = ( 5)( 9)
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