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Factoring Polynomials Finding GCF

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If you remember multiplication with real numbers, then you should remember the following facts: If 6 2 = 12, then we know 12 is the product of 6 and 2 6 and 2 are factors, or divisors, of 12. the quotient of 12 divided by 6 is 2. the quotient of 12 divided 2 is 6. So, to factor a number is to write it as the product of two or more numbers, usually natural numbers. Factoring and division are closely related.

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Review: a factor - is a number that is multiplied by another number to produce a product. A prime number - is any natural number greater than 1 whose only factors are 1 and itself. A composite number - is a number greater than 1 that has more than two factors. The prime factorization - is the factorization of a natural number that contains only prime numbers or powers of prime numbers.

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The figure below shows 24 square tiles arranged to form a rectangle. (4 tiles wide, 6 tiles long) Sketch other ways that the 24 tiles can be arranged to form a rectangle. ANSWER

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If you notice each rectangle has an area of 24, which includes (4 x 6, 3 x 8, 2 x 12, 1 x 24). Each of the numbers involved in these multiplications is a factor of 24. There are no other natural number pairs that have a product of 24, so 1, 2, 3, 4, 6, 8, 12, and 24 are the only factors of 24. Here are two examples

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Write 3 different factorizations of 16. Use natural numbers. 16 divided by Natural number168no4 2 1 x 16 2 x 8 4 x 4 8 x 2

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Finding Greatest Common Factor GCFs

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Lets make a list of factors of 36, written in order from least to greatest. 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 Make a list of factors for the number : 1, 2, 3, 6, 9, 18, 27, 54 Examine the two lists for factors that appear in both list. 36: 1, 2, 3, 6, 9, 12, 18, 36 54: 1, 2, 3, 6, 9, 18, 27, 54 common factors: 1, 2, 3, 6, 9, 18 GCF: 18

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What if the numbers in our previous example were expressions 36c 3 and 54c 2 ? 36c c c c c 2 18 c 2 54c c c c 2 18 c 2 GCF 18c 2 Try again

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Find the GCF: a) 18d 5 and 108d b) 18d and 5 c) 3m 3 n 3 and 9m 2 n 2 d) 4mn 3, 4m 2 n 3, and 16m 2 n 2

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Factoring a monomial from a polynomial Using GCF

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Factoring a Monomial from a Polynomial Factoring a polynomial reverses the multiplication process. To factor a monomial from a polynomial, first find the greatest common factor (GCF) of its terms. Find the GCF of the terms of: 4x x 2 – 8x List the prime factors of each term. 4x 3 = 2 · 2 · x · x x 12x 2 = 2 · 2 · 3 · x · x 8x = 2 · 2 · 2 · x The GCF is 2 · 2 · x or 4x.

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Find the GCF of the terms of each polynomial. a) 5v v 3 b) 3t 2 – 18 c) 4b 3 – 2b 2 – 6b d) 2x x 2 – 6x

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Use the GCF to factor each polynomial. a) 8x 2 – 12x b) 5d d c) 6m 3 – 12m 2 – 24m d) 4x 3 – 8x x Try to factor mentally by scanning the coefficients of each term to find the GCF. Next, scan for the least power of the variable.

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Factoring Out a Monomial Factor 3x 3 – 12x x Step 1 Find the GCF 3x 3 = 3 · x · x · x 12x 2 = 2 · 2 · 3 · x · x 15x = 3 · 5 · x The GCF is 3 · x or 3x Step 2 Factor out the GCF 3x 3 – 12x x = 3x(x 2 ) + 3x(-4x) + 3x(5) = 3x(x 2 – 4x + 5) To factor a polynomial completely, you must factor until there are no common factors other than 1.

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Factoring x 2 + bx + c Factoring x 2 + bx + c when c is positive

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Observe the two columns below, the multiplication of binomials on the left and the products on the right. What do you notice about the two list? 1.(x + 5)(x + 6) 2.(x + 3)(x + 10) 3.(x + 2)(x + 15) 4. (x + 1)(x + 30) [x x + 30] [x x + 30] [x x + 30] [x x + 30]

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The product of the constants = 30 The sum of the constants = the coefficient of the x-terms 1.(x + 5)(x + 6) 2.(x + 3)(x + 10) 3.(x + 2)(x + 15) 4.(x + 1)(x + 30) [x x + 30] [x x + 30] [x x + 30] [x x + 30] TRY THIS

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Remember the patterns you say earlier. x x + 30 Write the binomial multiplication that gives this product. ( )( ) What two constants multiplied together gives you 30? The sum of what two of those constants give you 17?

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In other words, to factor x 2 + bx + c, look for the factor pairs (the two numbers) whose product is c. Then choose the pair whose sum is b. Factor x 2 + 5x c = 6: write all (+ and -) the factor pairs of b = 5: choose the pair whose sum is = = = – 3 = Write the product using 2 and 3. Thus (x + 2)(x + 3) = x 2 + 5x + 6 a 2 + 9a + 20 TRY THIS NEXT

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Factor y 2 – 10y c = 24 Write all (+ and -) factor pairs of b = -10: choose the pair whose sum is = (-24) = = (-6) = Write the product using -4 and -6. (y – 4)(y – 6) = y 2 -10y + 24 TRY THIS n 2 -13n + 36

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Practice and Problem Solving. 1.t 2 + 7t + 10 = (t + 2)(t + ) 2.x 2 – 8x + 7 = (x – 1)(x - ) 3.x 2 + 9x + 18 = (x + 3)(x + ) Factor each expression 1.r 2 +4r n 2 – 3n k 2 + 5k x 2 – 2x y 2 + 6y + 8

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Factoring x 2 + bx + c Factoring x 2 + bx + c, when c is negative

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Find each product. What do you notice about the c term in each? (x – 5)(x + 6) (x – 3)(x + 10) (x – 2)(x + 15) (x – 1)( x + 30) (x + 5)(x - 6) (x + 3)(x - 10) (x + 2)(x – 15) (x + 1)(x – 30) x 2 + x – 30 x 2 + 7x – 30 x x – 30 x x – 30 x 2 – x – 30 x 2 – 7x – 30 x 2 – 13x – 30 x 2 – 29x - 30 What do you notice about c in each expression? What do you notice about c and the coefficient of the x-term?

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You can also factor x 2 + bx – c Factor x 2 + x – c = -20: write all (+ and -) factors of -20. (-1) 20 1 (-20) (-2) 10 2 (-10) (-4) 5 4 (-5) 2. b = 1: find the pair whose sum is 1. (-1) 20 1 (-20) (-2) 10 2 (-10) (-4) 5 4 (-5) 3. Write the product using -4 and 5 = 1 (x – 4)(x + 5) = x 2 + x - 20 TRY THIS n 2 + 3n - 40

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Factor z 2 – 4z – c = -12: write all (+ and -)factors of -12. (-1) 12 1 (-12) (-2) 6 2 (-6) (-3) 4 3 (-4) 2. b = -4: find the factor pair of -4. (-1) 12 1 (-12) (-2) 6 2 (-6) (-3) 4 3 (-4) 3. write the product using 2 and -6. (x + 2)(x – 6) = x 2 – 4z - 12 TRY THIS n 2 – 3n - 40

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Not every polynomial of the form x 2 + bx + c is factorable. Factor x 2 + 3x – 1 c = -1: the only factors of -1, are 1 and -1. b = 3: because , x 2 + 3x – 1 cannot be factored TRY THIS t 2 + 5t - 8

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Factor: 1. y y – x 2 – x – b 2 – 17b – s 2 + 4s – 5 5. y 2 + 2y – 63

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Factoring ax 2 + bx + c Factoring ax 2 + bx + c, when c is positive

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Before we tackle this factoring lets go back and review the F O I L method of how the product of two binomials works. (2x + 3)(5x + 4) = 10x 2 + (8x +15x) + 12 = 10x x + 12 First Terms Outer Terms Inner Terms Last Terms Notice what happens when the multiply the Outer Terms and Inner Terms. NEXT

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10x x + 12 To factor it, think of 23x as 8x + 15x. 8x + 15x = 23x 10x x + 12 = 10x 2 + 8x + 15x + 12 Where did we get 8x and 15x? Notice that multiplying (a) 10 and (c) 12 gives you 120, which is the product of the x 2 -coefficient (10) and the constant term (12). 8 and 15 are factors of also 8x + 15x = 23 This example suggest that, to factor a trinomial, you should look for factors of the product ac that have a sum of b. Lets see if it works. In the form of ax 2 + bx + c

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Consider the trinomial 6x x + 7. To factor it, think of 23x as 2x + 21x. Where did we get 2 and 21? If we multiply 6 and 7 we get 42, which is the product of the x 2 -coefficient (6) and the constant (7). 2 and 21 are factor of and 2x + 21x = 23x Yes it does work! So we must find the product of ac that have the sum b. NEXT

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Now we must rewrite the trinomial using the factors you found for b. (2 and 21) 6x 2 + 2x + 21x + 7 Now we are going to find the GCF by grouping terms. (Remember the Associative Property) (6x 2 + 2x) + (21x + 7) 2x(3x + 1) + 7(3x + 1) Now lets use the Distributive Property to write the two binomials. (2x + 7)(3x + 1) FACTOR What do you notice about the terms in the parenthesis? Now we have factored 6x x + 7 to (2x + 7)(3x + 1)

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Factor 5x x + 2 5x x + 2 = 5x 2 + 1x + 10x + 2 Rewrite bx: 11x = 1x + 10x. = (5x 2 + 1x)(10x + 2) Group terms, Associative Property. = x(5x + 1) + 2(5x + 1) Factor GCF of each pair of terms. = (x + 2)(5x + 1) Use Distributive Property to write factored terms. Step 1: Find factors of ac that have a sum b. Factors of 101 x 102 x 5 Sums of factors117 Since ac = 10 and b = 11, find the positive factors of 10 that have a sum 11. Step2: To factor the trinomial, use the factors you found (1 + 10) to rewrite bx. TRY ANOTHER 6x x + 5

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What is the factored form of 6x x + 5? 6x x + 5 = 6x 2 + 3x + 10x + 5 Rewrite bx: 13x = 3x + 10x. (6x 2 + 3x)(10x + 5) Group terms together to factor. 3x(2x + 1) + 5(2x + 1) Factor GCF of each pair of terms. (3x + 5)(2x + 1) Use Distributive Property to write binomials.

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Factor: 1) 2n n + 5 2) 5x x ) 2y 2 – 23y ) 4y y ) 8t t + 15

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Factoring ax 2 + bx +c Factoring when ac is negative

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The sums of positive and negative numbers gives us our b. Can we apply the same steps we have learned to factor trinomials that contain negative numbers? Yes. Your goal is still to find factors of ac that have sum b. Because ac < 0 (less than), the factors must have different signs. We need to use all combinations of factors. Ex: factors of (-15) (-1)15 3(-5) (-3)5 1+(-15)=-14 (-1)+15=14 3+(-5)=-2 (-3)+5 =2

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Factors of -451, -45-1, 453, -15-3, 155, -9-5, 9 Sums of factors Factor 3x 2 + 4x – 15 Find factors of ac with the sum b. Since ac = -45 and b = 4, find factors of x 2 -5x +9x – 15 Rewrite bx: 4x = -5x + 9x. (3x 2 -5x) + (9x – 15) Group terms together to factor. x(3x – 5) + 3(3x– 5) Factor GCF of each pair of terms. (x + 3)(3x – 5) Use Distributive to rewrite binomials.

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Not all expressions of the form ax 2 + bx – c can be factored. This is especially common when the polynomial contains subtraction. Try this. -10x x - 5 Factor: 1) 3k 2 + 4k – 4 2) 5x 2 + 4x – 1 3) 10y 2 – 11y – 6 4) 6q 2 – 7q – 49 5) 2y y – 90 TRY THIS

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Geometry The area of a rectangle is 2y 2 – 13y – 7. What are the possible dimensions of the rectangle? Use factoring. 2y 2 – 13y – 7 =

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Simplifying before factoring Some polynomials can be factored repeatedly.

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Some polynomials can be factored repeatedly. This means you can continue the process of factoring until there are no common factors other than 1. If a trinomial has a common monomial factor, factor it out before trying to find binomial factors. Take for example: (6h + 2)(h + 5) and (3h + 1)(2h + 10) Find each product. 6h h + 10 and 6h+ 32h + 10 What do you notice about these two polynomials? Can you factor a monomial before factoring a binomial? NEXT

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Step 2: Rewrite bx using factors. Factor 20x x + 35 completely. 20x x + 35 = Factor out GCF monomial. 5(4x x + 7) = Rewrite bx: 16x = 2x + 14x. 5[4x 2 + 2x + 14x + 7)] = Factor GCF of each pair of terms. 5[2x(2x + 1) + 7(2x + 1)] = 5(2x + 7)(2x + 1) Rewrite using the Distributive Property. Step 1: Find factors of ac with sum b. Factors of 281 x 282 x 144 x 7 Sum of factors291611

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Solving Polynomial Equations by Factoring Finding x

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If you remember graphing linear equations, many times the line crossed the x-axis. You could find the x- and y-intercepts of these lines. Polynomials have the same characteristics, but quadratics can have no x-intercept, one x- intercept or two x-intercepts. x-intercept y-intercept x-intercept y-intercept

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Make a table of the polynomial shown below. y = x 2 + 2x – 3 Identify the numbers that appear to be the x-intercepts. Rewrite the equation by factoring the right side. [y = (x + 3)(x – 1)] If you notice the x-intercept are solutions to the equation (x + 3)(x – 1) = 0 x-intercept = -3 x-intercept = 1

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Remember the Standard Form of a Quadratic Equation ax 2 + bx + c, where a 0. The value of the variable in a standard form equation is called the solution, or the root, of the equation. Lets discuss the Multiplication Property of Zero which states that if a = 0 or b = 0, then ab = 0. We can use the Zero-Product Property to solve quadratic equations once the quadratic expression has been factored into a product of two linear factors. Lets see how we can use this property to solve quadratic expressions.

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Apply the Zero- Product Property. Solve (4x + 5)(3x – 2) = 0 (4x + 5)(3x – 2) = 0 The quadratic expression has already been factored. 4x + 5 = 0 or 3x – 2 = 0 If (4x + 5)(x – 2) = 0, then (4x + 5) = 0 or 3x – 2) = 0. 4x = -5 3x = 2 x = -5/4 x = 2/3 Solve each equation for x. x-intercept TRY THIS

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Solve (x + 5)(2x – 6) = 0 (x + 5)(2x – 6) = 0 x + 5 = 0 or 2x – 6 = 0 Use the Zero-Product Property. 2x = 6 Solve for x. x = -5 or x = 3 Substitute – 5 for x.Substitute 3 for x. (x + 5)(2x – 6) = 0(x + 5)(2x – 6) = 0 (-5 + 5)[2(-5) – 6] = 0(3 + 5)[2(3) – 6) = 0 (0)(-16) = 0 (8)(0) = 0

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Solve 1) (x + 7)(x – 4) = 0 2) (x – 3)(x – 7) = 0 3) (x + 4)(2x – 9) = 0 4) (2x + 3)(x – 4) = 0 5) (x + 3)(x + 5) = 0

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You can also use the Zero-Product Property to solve equations of the form ax 2 + bx + c = 0, if the quadratic expression ax 2 + bx + c can be factored. Solve x 2 – 8x – 48 = 0 (x – 12)(x + 4) = 0 Factor x 2 – 8x – 48. x – 12 = 0 or x + 4 = 0 Use the Zero-Product Property. x = 12 or x = -4 Solve for x. Try This x 2 + x – 12 = 0

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Before solving a quadratic equation, you may need to add or subtract terms in order to write the equation in standard form. Then factor the quadratic expression. Solve 2x 2 – 5x = 88 2x 2 – 5x – 88 = 0 Subtract 88 from each side. (2x + 11)(x – 8) = 0 Factor 2x 2 – 5x – 88. 2x + 11 = 0 or x – 8 = 0 Use the Zero=Product Property. 2x = -11 or x = -8 Solve for x. x = -5.5 Try This x 2 – 12x = -36

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Solve 1) b 2 + 3b – 4 = 0 2) y 2 – 3y – 10 = 0 3) 2z 2 – 10z = -12 4) n 2 + n – 12 = 0 5) x 2 + 8x = -15 6) 4y 2 = 25 MORE

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Write each equation in standard form. Then solve. 1) 2q q = -60 2) 3a 2 + 4a = 2a 2 – 2a – 9 3) 4x = 10x + 3x 2 – 4 4) 3t 2 + 8t = t 2 – 3t – 12

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Solving Cubic Equations by Factoring Cubic Equations ax 3 +bx 2 + cx + d = 0

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The standard form of a cubic equation in x is any equation that can be written in the form ax 3 + bx 2 + cx + d = 0, where a 0. Using an extension of the Zero-Product Property, you can solve many cubic equations. If a, b, and c represent real numbers and abc = 0, [(5)(4)(0) = 0] then a = 0, b = 0 or c = 0. For example, if x(x – 2)(3x + 4) = 0, you can write the following. x = 0 or x – 2 = 0 or 3x + 4 = 0 Solving each for x, then x = 0, 2, and -4/3

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Solve 2n 3 + 8n 2 – 42n = 0 2n 3 + 8n 2 – 42n = 0 Factor the GCF, 2n, from each term. 2n(n 2 + 4n – 21) = 0 Factor n 2 + 4n – 21. 2n(n + 7)(n – 3) = 0 Apply the Zero-Product Property. 2n = 0 or (n + 7) = 0 or (n – 3) = 0 Solve for n. n = 0 or n = - 7 or n = 3 The solutions are 0, -7, and 3. Try 10k 3 – 13k 2 + 4k =0

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Solve m m m = 0 m m m = 0 Factor for GCF, m, from each term. m(m m + 121) = 0 Factor m m m(m + 11)(m + 11) = 0 Apply the Zero-Product Property. m = 0 or m + 11 = 0 or m + 11 = 0 Solve for n. m = 0 m = -11 m = - 11 The solution are 0 and -11. Try z 3 – 14z z = 0

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The steps for solving a polynomial by factoring are: Step 1: Write the equation in standard form. Step 2: Factor the GCF, if one exists, from each term in the equation. Step 3: Factor the polynomial Step 4: Apply the Zero-Product Property and set each factor equal to zero. Step 5: Solve for the variable. Step 6: Check you solution(s) in the original equation.

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