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Factor by Splitting the Middle Term

Factor by Splitting the Middle Term. Grade 10. What to look for:. t rinomial (3 terms) written in ax² + bx + c, when a > 1 Examples: 4x² + 12x + 32 … a=4 6x² - 14x + 48 … a=6. Why do some polynomials have 3 terms and others have 4? When the FOIL multiplication method

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Factor by Splitting the Middle Term

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  1. Factor by Splitting the Middle Term Grade 10

  2. What to look for: • trinomial (3 terms) • written in ax² + bx + c, when a > 1 Examples: 4x² + 12x + 32 … a=4 6x² - 14x + 48 … a=6

  3. Why do some polynomials have 3 terms and others have 4? When the FOIL multiplication method is used to multiply2 binomials, for example: (x + 3)(4x - 2) or (a + b)(c - d) If the Outer and Inner terms can be combined, the result will be a Trinomial (3 terms). (x + 3)(4x - 2)= 4x2- 2x + 12x - 6 = 4x2+ 10x - 6 If they cannot be combined, it keeps all 4 terms. (a + b)(c - d) = ac - ad + bc - bd

  4. First (a) Middle (b) Last (c) 3 terms: 2x2+ 9x + 10 II. Factoring a Trinomial of the form: ax2 + bx + c First, we will split the middle term into the original outer and inner terms. This gives us a four term polynomial which we can factor using the grouping method.

  5. first last middle Step 1: Multiply first*last 2 x 10 = 20 2x2+ 9x + 10 20 Step 3: The sign of the last term tells us whether to add or subtract the factors of 20. Add Step 4: Which pair of factors gives us the middle term? Splitting the middle term Step 2: Find all the FACTORS of 20 1  20 2  10 4  5 = 21 = 12 = 9 Since the middle term is 9x, the original 4 and 5 each had an“x”. (They were like terms and were added together) Step 5: Choose the correct signs. FACT: We add numbers when they have the same signs. (step 3). i.e. The factors are both positive, + 4x and + 5x ORboth negative, -4x and -5x + 4x and + 5x = + 9x This pair gives us the correct middle term.

  6. 2x2+ 9x + 10 Note: We did step 1 on the previous page 1. Split the middle term into 2 terms 2x2+ 4x+ 5x + 10 2. Factor by grouping (2x2+ 4x)+ (5x+ 10) 2x(x + 2 )+ 5(x + 2) (x + 2 )+ (2x + 5) HCF Answer: (x + 2)(2x + 5)

  7. 24 SUBTRACT This pair works correct terms: -12x and +2x Subtraction example first last Step 1: Multiply first*last 3x2- 10x- 8 3 x 8 = 24 Step 2. Find all the factors of 24 1 * 24 2 * 12 3 * 8 4 * 6 = 23 = 10 = 5 = 2 Step 3: Pick the pair that subtract to equal -10x (the middle term) Step 4: Pick the correct signs: (Subtract means: different signs) +12x - 2x = +10x - 12x + 2x = -10x

  8. 3x2+ 10x - 8 Note: We did step 1 on the previous page 1. Split the middle term into 2 terms 3x2+ 2x- 12x - 8 2. Factor by grouping (3x2- 12x ) + (2x - 8) 3x(x - 4 ) + 2(x - 4) (x - 4 )+ (3x + 2) HCF Answer: (x - 4)(3x + 2)

  9. SPECIAL CASE!!! There are some polynomials that have a factor that must be taken out beforeusing the box. Take a polynomial such as: 4x2 + 10x - 6 This polynomial has a common factor of “2” in all three terms. If this is not taken out before using the box, the “2” will be taken out twice, doubling the answer. (Example on the next two slides)

  10. The way this problem SHOULD be worked - Step 3: Subtract 1 * 6 2 * 3 This problem has a GCF of “2”. We’ll factor it out then work the problem normally using only the trinomial (in parentheses). 4x2 + 10x - 6 = 2 (2x2 + 5x - 3) Step 1: Multiply the first*last 2 * 3 = 6 Step 2: Find the factors of 6 = 5 = 1 Step 4: Choose the pair of factors that equal + 5x (the middle term) Step 5: Choose the correct signs: -x and +6x =+5x or x and -6x = -5x

  11. 2(2x2+ 5x - 30 Note: We did step 1 on the previous page 1. Split the middle term into 2 terms (2x2+ 6x- x – 3) 2. Factor by grouping (2x2+ 6x )- (x+ 3) 2x(x + 3 ) - (x + 3) (x + 3 )+ (2x - 1) Don’t forget to put that extra “2” in the answer!!! HCF Answer: 2(x + 3)(2x - 1)

  12. Some practice problems Answers: 1. (4x + 3)(x - 2) 2. (3y - 1)(y - 5) 3.(2x - 3)(x + 6) 4. (x + 1)(x + 2) 5. (x - 6)(x + 2) 4x2 - 5x - 6 1. 2. 3. 3y2 - 16y + 5 2x2 + 9x - 18 4. x2 + 3x + 2 5. x2 - 4x - 12

  13. Resource www.greenbox.com

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