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Polynomial Review

Polynomial Review. What is a polynomial? An algebraic expression consisting of one or more summed terms, each term consisting of a coefficient and one or more variables raised to natural number exponent. Examples: X 2 + 3x – 3 2x + 1 5y 3 + 2y 2 - 4y - 8. Terms.

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Polynomial Review

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  1. Polynomial Review • What is a polynomial? • An algebraic expression consisting of one or more summed terms, each term consisting of a coefficient and one or more variables raised to natural number exponent. • Examples: • X2 + 3x – 3 • 2x + 1 • 5y3 + 2y2 - 4y - 8

  2. Terms • What is a term? • A term is the product of a coefficient and one or more variables raised to a natural number exponent • Examples: • X • 2xy • 3y3 x • 4

  3. Terms cont. • Identify the coefficient and degree for each example • X • Coefficient – 1 Degree – 1 • 2xy • Coefficient – 2 Degree – 2 • 3y3 x • Coefficient – 3 Degree – 4 • 4 • Coefficient – 4 Degree - 0

  4. Types of Polynomials • Monomial • Has 1 term 3ab 2y3 xy • Binomial • Has 2 terms combined with addition or subtraction ax2 + bx 2x + 3 • Trinomial • Has 3 terms combined with addition or subtraction ax2 + bx + c 2x + 3x +1

  5. Multiplying Polynomials • Binomial x Binomial – FOIL Method • First Outer Inner Last F – x * x O – x * 2 I – 1 * x L – 1 * 2

  6. FOIL cont. • F: x * x = x2 • O: x * 2 = 2x • I: 1 * x = x • L: 1 * 2 = 2 • Add the terms: • x2 + 2x + x + 2 • Combine like terms: • x2 + 3x + 2

  7. Chapter 6 – Factoring Polynomials and Solving Equations • 6.1 Introduction To Factoring

  8. 6.1 Introduction To Factoring • Factors – 2 or more numbers that • multiply to a new number • Factors of 4: 1,4 and 2,2 • 1 * 4 = 4 and 2 * 2 = 4 • Factors of 32: 1, 32 2, 16 4, 8 • 1 * 32 = 32 2 * 16 = 32 • and 4 * 8 = 32

  9. 6.1 Introduction To Factoring • Factors of a polynomial – 2 or more • polynomials that multiply to a higher • degree polynomial • x2 – x • x2 and x have a common factor of x • x2 = x * x • x = 1 * x • We can use the distributive property to pull the common factor out of each term. • x(x – 1)

  10. 6.1 Introduction To Factoring • Find the common factors of the • following terms: • 8x2 + 6x • 8x2 = 2x * 4x • 6x = 2x * 3 • The common factor is 2x • Pull out the common factor using the distributive property. • 2x(4x + 3)

  11. 6.1 Introduction To Factoring • 10x + 6 • Step 1: Find the common factors for the terms • Common factor: 2 • Step 2: Pull the common factor out from each term • 10x = 2 * 5x • 6 = 2 * 3 • Step 3: Multiply the common factor by the sum of the other factors • 2(5x + 3)

  12. 6.1 Introduction To Factoring • 6x2 + 15x • Step 1: Find the common factors for the terms • Common factor: 3x • Step 2: Pull the common factor out from each term • 6x2 = 3x * 2x • 15x = 3x * 5 • Step 3: Multiply the common factor by the sum of the other factors • 3x(2x + 5)

  13. 6.1 Introduction To Factoring • Factor the polynomial: 3z3 + 9z2 – 6z • Find the common factor: 3z • Pull out the common factor: z2 +3z – 2 • Write the polynomial as the product of the factors • 3z (z2 + 3z – 2)

  14. 6.1 Introduction To Factoring • Factor the polynomial: 2x2y2 + 4xy3 • Find the common factor: 2xy2 • Pull out the common factor: x + 2y • Write the polynomial as the product of the factors: • 2xy2(x + 2y)

  15. 6.1 Introduction To Factoring • Find the GCF and write each polynomial as the product of the factors. • 9x2 + 6x • 8a2b3 – 16a3b2 • 66t + 16t2

  16. 6.1 Introduction To Factoring • Find the GCF and write each polynomial as the product of the factors. • 9x2 + 6x • 3x (3x + 2) • 8a2b3 – 16a3b2 • 8a2b2(b - 2a) • 66t + 16t2 • 2t(33 + 8t)

  17. 6.1 Introduction To Factoring • Factoring by Grouping • What is common between the following terms? • 5x(x + 3) + 6(x+ 3) • The common factor is the binomial (x + 3) • We can rewrite the polynomial as the product of the common factor and the sum of the other factors: • (x + 3)(5x + 6)

  18. 6.1 Introduction To Factoring • Identify the common factors • x2(2x – 5) – 4x(2x – 5) • 5x(3x – 2) + 2(3x – 2) • (z + 5)z + (z + 5)4

  19. 6.1 Introduction To Factoring • Identify the common factors • x2(2x – 5) – 4x(2x – 5) • 2x – 5 • 5x(3x – 2) + 2(3x – 2) • 3x – 2 • (z + 5)z + (z + 5)4 • z + 5

  20. 6.1 Introduction To Factoring • Factor by grouping: 2x3 – 4x2 + 3x – 6 • Step 1: Group the terms that have common factors (2x3 – 4x2) + (3x – 6) • Step 2: Identify the common factors for each group • Common factor of 2x3 – 4x2 : 2x2 • Common factor of 3x – 6: 3

  21. 2x3 – 4x2 + 3x – 6 cont. • Step 3: write each grouping as the product of the factors • 2x2(x – 2) + 3(x – 2) • Note: the parenthesis are the same • Step 4: Distribute the common factor from each grouping • 2x2(x – 2) + 3(x – 2) • (x – 2)(2x2 + 3)

  22. 6.1 Introduction To Factoring • Factor by Grouping: 3x + 3y + ax + ay • (3x + 3y) + (ax + ay) • 3(x + y) + a(x + y) • (x + y) (3 + a) • Check by using FOIL: (x + y) (3 + a) • 3x + ay + 3y + ay

  23. 3x + 3y + ax + ay cont. • Let's Factor this polynomial again by • grouping the x terms and y terms • 3x + 3y + ax + ay = 3x + ax + 3y + ay • (3x + ax) + (3y + ay) • x(3 + a) + y(3 + a) • (3 + a) (x + y)

  24. 3x + 3y + ax + ay cont. • Compare Grouping 1 and Grouping 2 • Grouping 1: (x + y) (3 + a) • Grouping 2: (3 + a) (x + y) • Are these the same?

  25. Reflect • Does it matter how you group the terms? • No, you will get the same answer • However, you need to group terms that have a common factor other than 1.

  26. Reflect • Do the parenthesis have to be the same • for each term after you have factored • the GCF? • Yes • How many terms do you need to factor • by grouping? • At least 4

  27. Practice • Factor the following Polynomials • 6x3 – 12x2 – 3x + 6 • 2x5 – 8x4 + 6x3 – 24x2 • 4t3 – 12t2 + 3t – 9 • Check by multiplying the factors

  28. Practice cont. • Answers: • 3(2x2 – 1)(x – 2) • 2x2 (x2 + 3)(x – 4) • (4t2 + 3)(t – 3)

  29. ???? Questions ????

  30. 6.2 Factoring Trinomials – x2 + bx + c

  31. 6.2 Factoring Trinomials – x2 + bx + c • When factoring a polynomial in the • form of x2 + bx + c, we will be • reversing the FOIL process • So, let's review FOIL

  32. FOIL • F (x + m) (x + n) = x2

  33. FOIL • F (x + m) (x + n) = x2 • O (x + m) ( x + n) = nx

  34. FOIL • F (x + m) (x + n) = x2 • O (x + m) ( x + n) = nx • I (x + m) (x + n) = mx

  35. FOIL • F (x + m) (x + n) = x2 • O (x + m) ( x + n) = nx • I (x + m) (x + n) = mx • L (x + m) (x + n) = nm

  36. FOIL • F (x + m) (x + n) = x2 • O (x + m) ( x + n) = nx • I (x + m) (x + n) = mx • L (x + m) (x + n) = nm • x2 + nx + mx + nm

  37. FOIL • Can we combine like terms? • Yes, O and I in FOIL give us like terms • x2 + nx + mx + nm • What is the common factor? • X • Pull out the x: • x2 + (n + m)x + nm

  38. Factoring • Compare • (x +n)(x + m): • x2 + (n + m)x + nm • Let's • Standard Form: • x2 + bx + c

  39. Factoring • Based on the Standard Form: x2 + bx + c, • what is the b in the polynomial we found: • x2 + (n + m)x +nm • b = (n + m)

  40. Factoring • Based on the Standard Form: x2 + bx + c, • what is the c in the polynomial we found: • x2 + (n + m)x +nm • c = nm

  41. 6.2 Factoring Trinomials – x2 + bx + c • To factor the trinomial x2 + bx + c, • find two numbers m and n that • satisfy the following conditions: • m * n = c • m + n = b • Such that x2 + bx + c = (x + m)(x + n)

  42. 6.2 Factoring Trinomials – x2 + bx + c • Remember that (m + n)x was the • sum of the O and I in FOIL • and • nm was the L in FOIL.

  43. x2 + 7x + 12 • Step 1: Factor x2 • x2 + 7x + 12 • (x )(x )

  44. x2 + 7x + 12 • Step 2: Identify all of the factors of 12 • 1 * 12 = 12 • 2 * 6 = 12 • 3 * 4 = 12

  45. x2 + 7x + 12 • Step 3: Identify the factors of 12 • that add to 7 • 1 * 12 = 12 1 + 12 = 13 • 2 * 6 = 12 2 + 6 = 8 • 3 * 4 = 12 3 + 4 = 7

  46. x2 + 7x + 12 • Step 3: Identify the factors of 12 • that add to 7 • 1 * 12 = 12 1 + 12 = 13 • 2 * 6 = 12 2 + 6 = 8 • 3 * 4 = 12 3 + 4 = 7

  47. x2 + 7x + 12 • Step 4: Complete the factors • (x + 3)(x + 4)

  48. x2 + 7x + 12 • Step 5: Check by FOIL • (x + 3)(x + 4) • x2 + 4x + 3x + 12 • x2 + 7x +12

  49. x2 + 13x + 30 • Step 1: Factor x2 • x2 + 13x + 30 • (x )(x )

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