Factoring and Applications


 Agnes Skinner
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1 Factoring and Applications What is a factor? The Greatest Common Factor (GCF) To factor a number means to write it as a product (multiplication). Therefore, in the problem 48 3, 4 and 8 are called the factors and 3 is the product. The GCF (greatest common factor) of a list of numbers is the largest factor that is common to all numbers in the list. For example, the GCF of 16 and 0 is 4. One way to find the GCF is by inspection. Another way to find the GCF is to use prime factorization. Write the prime factorization of each number in the list in exponential form. Circle the common factors in the list. The GCF is the product of the common factors, each raised to the lowest power that appears. Find the GCF of 45, 75 and The GCF is 3515 To find the GCF for a list of variable terms, include in the GCF all variables that are in common to all terms, raised to the lowest power that appears. In the example below, we do not include y in the GCF as it does not appear in the first term. 3 4 Find the GCF for 10 x, 0x y 3 The GCF is 10x If all terms in the list are negative, it is common to factor out 1 with the GCF. To find the GCF of a polynomial, find the GCF of all terms in the polynomial. For example: 3 Find the GCF of 4x 8x 1xy The GCF is 4x To factor out the GCF from a polynomial may be thought as undoing the distributive property. The distributive property tells us how to distribute the operation of multiplication over addition/subtraction. 3(5x7) 15x 1. To factor out the GCF reverses this process: 15x1 3(5x 7) To factor the GCF out of a polynomial, first find the GCF of all terms. Rewrite each term as the product of the GCF and the remaining factors. Write the GCF in front of the parentheses. Inside the parentheses should be the remaining factors, and same number of terms as the original polynomial. If one of the terms is equal to the entire GCF, then include a 1 or a 1 in the parentheses for that term. 3 6x 1x x x(3x 6x 1) When a polynomial has four terms, try factoring by grouping.
2 To factor by grouping, group the first two terms together and the last two terms together. Find the GCF for the first terms and factor, and then find the GCF for the last terms and factor. Then, factor the common binomial term out of the polynomial. ab 3a b 6 4 terms? Try grouping ( ab 3 a) ( b 6) Group together first two and last two terms. a( b 3) ( b 3) For each group, find the GCF and factor. ( b 3 )( a ) Factor out the common binomial factor of ( b 3). Remember that order does not matter when multiplying. This is also known as the commutative property of multiplication. Therefore, in the last example, the answer may be written as ( b3)( a ) or ( a)( b 3). Both forms are correct. 1. Find the GCF for the list of numbers. 8,16, ab, 75 ab,100a b 3,5,15 x y z. Factor out the GCF from the polynomial. 3 6t 18t x 0x 10x ab ab x(5y3) 9(5y 3) 3. Factor out a negative greatest common factor if the lead coefficient of the highest degree term is negative. x 6x 7 5p 15q 3 1y 6y 4y 4. Factor the polynomial by grouping. p 5pqpq 10q x 7x8x 8 a ab6ab 3b gh h 6g 3 How to factor a trinomial with lead coefficient of 1 Recall that a trinomial is a polynomial with 3 terms. For example, x 6x 5 is a trinomial. To factor a trinomial into the product of two binomials, may be thought of as reversing the FOIL process. To multiply two binomials using FOIL looks like:
3 ( x3)( x7) x 7x3x1 x 10x1 To factor a trinomial is the process to reverse FOIL and looks like: Trinomial Factored Form x 10x1 ( x3)( x7) Recall the lead coefficient is the coefficient of the highest degree term. The lead coefficient of x 7x9is 1 and the lead coefficient of 5x x is 5. To factor a trinomial with lead coefficient = 1: x bx c Always factor out the GCF first. List the factors of Find the pair that adds to Rewrite the trinomial as: ( x )( x ), where the blank terms in each binomial are the pair from part To check, multiply out the factored form using FOIL. If no such pair of integers can be found in the steps in part b above, then answer that the polynomial is prime. This means it cannot be factored over the integers. If the instructions say to factor completely, always look for the GCF first. 1. Factor each trinomial. x 1x 35 y y 6 a 3a 54 b 1b 3 e. x 1x 10. Factor each polynomial completely. 3x 1x 36 3 x x 48x x 6x 5 How to factor a trinomial with lead coefficient other than 1 There are two ways commonly taught to factor ax bx c. The first way is either called guess and check or factoring using FOIL. This is best done by demonstration, so check your class notes and book if this is how your instructor does it. The second way is known either as the ac method or factoring a trinomial by grouping. The steps will be listed below, The ac method to factor: ax bx c
4 Find the values of a, b, and Multiply a c. List all the factors of a Find the pair that adds to Rewrite the middle bx term as the sum of two terms with coefficients from part e. Factor by grouping. 1. Factor: x 7x 3 3x 10x 8 10x 9x 4x x 15. Factor completely: 3 6x 33x 15x 6x 11x 4 5a 55ab 10b x y 4x y 5xy 3 Factoring Special Forms How to factor a difference of squares: A B ( AB)( A B). Be very carefully to note that a sum of squares may only be factored if it has a GCF. It cannot be factored as a difference of squares. Perfect square trinomials can be factored using a special pattern. Ask your instructor if they want you to know this pattern! Otherwise, you can factor perfect square trinomials using the ac metho Perfect Square Trinomial Factoring: A AB B ( A B) A AB B ( A B) Summary of Factoring Techniques: when you are facing a factoring problem, use the following steps: 1. Always factor out the GCF first, if there is one.. 4 terms? Try grouping terms? Use the ac method to factor the trinomial. You can also look to see if it is a perfect square trinomial. 4. terms? Is it a difference of squares? A B ( AB)( A B)
5 1. Factor: x y 1 a 9 5a 36b e. y 4. Factor: a 4a 4 1x 1x 9 9y 1y 4 x 14x Examine the student s work below. What did the student do wrong? x 16 ( x4)( x 4) 4. Examine the student s work below. What did the student do wrong? t t1 ( t 1) Solving Quadratic Equations by Factoring A quadratic equation can be written in the form: ax bx c 0 The zero product property: If ab 0 then a0 or b 0 Quadratic equations can have 0, 1 or solutions. To solve a quadratic equation by factoring, use the following steps: Write the equation in standard form: ax bx c 0. This means the right side must be equal to 0 and the exponents are in descending order. Factor the left side completely. Use the factoring techniques you have learned in class. Use the zero product property to set each factor equal to 0. Solve each equation from step c for x. e. Check the solution by substituting into the original equation. 1. Solve each equation. Check the solution(s). x 5x6 0 t 64 0
6 6a a 0 9x 30x 5 Applications of Quadratic Equations Ask your instructor which geometry formulas you need to know. Commonly, you will be required to know the area of a square, area of a rectangle, area of a triangle, and the Pythagorean Theorem. Area of a square with side length s: A s Area of a rectangle with sides of length l and w: Alw 1 Area of a triangle with base b and height h: A bh In a right triangle with legs of length a and b, and hypotenuse of length c: a b c Consecutive integers mean one integer right after another integer. If the first integer is n, then the second integer is n+1. Consecutive odd or even integers are numbers apart. If the first integer is n, then the second integer is n+. 1. The sum of two consecutive odd integers is 108. Find the two integers.. The product of two consecutive positive integers is 110. Find the two integers. 3. The length of a rectangle is 4 meters more than the width. The area of the rectangle is 96m. Find the length and the width of the rectangle. 4. The height h of a ball in feet t seconds after being thrown from a roof can be estimated by the equation h 16t 64t 80. Find the time when the ball will hit the groun Hint: when the ball hits the ground, its height is 0 feet.
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