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By: Jaclyn Hart

By: Jaclyn Hart. All About Polynomials. What is a Polynomial?. pol -uh- noh - mee -uh l. Noun: An expression including the sum of two or more terms each of which is the product (multiplication) of a constant and a variable raised to an integral (exponent) power:

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By: Jaclyn Hart

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  1. By: Jaclyn Hart All About Polynomials

  2. What is a Polynomial? pol-uh-noh-mee-uh l Noun: An expression including the sum of two or more terms each of which is the product (multiplication) of a constant and a variable raised to an integral (exponent) power: ax2 + bx+ cis a polynomial, where a, b, c are constants and xis a variable.

  3. Term Coefficient Degree What do they all mean??

  4. Terms 6x2 + 4x - 9 A term is: any number, variable, product of the two

  5. 6x2 + 4x - 9 Constant 9 is a constant term because it only contains a number, and has no variable to change it’s value.

  6. 6x2 + 4x - 9 Coefficient: the numeric factor of the term so if a term has a number and a variable, the coefficient is the “number” *in other words it is the number in front of the variable* 6x2 4x The coefficients from above are: Number Variable Number Variable Coefficients

  7. Finding the degree of a Polynomial The sum of exponents for variables Degree of a term EX 2x has an exponent value of 1 therefore it has a degree of 1 6x2y4 has exponent values of 2 & 4. so add 2 + 4 = 6 therefore, degree is 6 Not this kind of degree Exponent on X Exponent of Y

  8. Degree of a Polynomial • The degree of a polynomial is determined by the largest degree of all terms • Found easiest if terms are written in descending order • This would put the degree of the polynomial as the first term written • 4x4 + 8x3 + 2x2 + x – 4 descending order

  9. Now you give it a shot Practice with these examples by putting them in descending order. Click on button to see correct answer 4x3 – 18x6 + 3x2 – 8 + 12x4 3z2 + 6z3 - 21z4 – 8z 12a – 3a2 + 13a4 – a3 – 2 9b2 + 10b3 – 2 + b Answer Answer Answer Answer

  10. 4x3 – 18x6 + 3x2 – 8 + 12x4 – 18x6 + 12x4 + 4x3 + 3x2 - 8 Since there are no terms with a 5th degree or a degree of 1 we do not right anything in there place and we move to the next degree

  11. 3z2 + 6z3 - 21z4 - 8z - 21z4 + 6z3 + 3z2 - 8z

  12. 12a - 3a2 + 13a4 - a3 - 2 13a4 - a3 - 3a2 + 12a - 2

  13. 9b2 + 10b3 – 2 + b 10b3 + 9b2 + b - 2

  14. Time to master the different types of polynomials Click on each box to learn Monomial Binomial Trinomial

  15. Monomials A monomial is a polynomial with ONE term 5x 2ab 6x2 7z2 8

  16. Binomials A polynomial with TWO terms 5a – 3x2 16z + 16y by – ay 4c – d 2z3 + 10 13k + h4

  17. Like a tricycle Trinomial 7z3 - 6z2 + 2 4 – 8x + 4y A polynomial with THREE terms 9y – 3y2 + y5 3x2 + 4x3 + x

  18. “Like” terms 3x2 + x3 NOT like terms Like terms are terms That have the exact Variables including exponents 3x2 + x2 Like terms

  19. Its time to combine “like” terms • Adding and Subtracting • You can only combine terms that are “like” • Let’s start with something easy • 4x + x = 5x • 3x2 + 6x2 = 9x2 • 7y2 + 8y = careful you can’t combine these

  20. Adding & subtracting • To make it easier put like terms together 4x2 + 5y2 – 2x2 + 5y2 (4x2 – 2x2) + (5y2 + 5y2) • Now combine 2x2 + 10y2 TRY ANOTHER: 12y3 + 3y5 – 2y5 + 2y3 = (12y3 + 2y3) + (3y5 – 2y5) = 14y3 + y5

  21. Adding and Subtracting with Parenthesis If there is a + in front of the parenthesis the terms stay the same • Now that you know the basics here are some tricky ones However if there is a – in front of the parenthesis you must distribute it to all the terms inside the parenthesis To distribute multiply all terms by -1 Here are some examples click on example to see step by step instructions (3x2 – 4y + 2z) – (2y – 8z + 8x2 ) (4z – 8y2 ) – ( -4y2 - 6z) (12a – b + 6ab) - (11ab – 2ab + 7a) (2c – 2c2 + c3) – (5c + 3c3 – 2c2)

  22. Distribute -1 (3x2 – 4y + 2z) – (2y – 8z + 8x2 ) Keep the same = 3x2 - 4y + 2z – 2y + 8z – 8x2 2 negatives (-) make a positive Match up “like” terms (3x2 – 8x2 ) + (-4y – 2y) + (2z + 8z) Combine (-5x2 ) + (-6y) + (10z) * Be careful of signs Remove parenthesis -5x2 - 6y + 10z

  23. Distribute (4z – 8y2 ) – ( -4y2 - 6z) 4z – 8y2 + 4y2 + 6z Pair up (4z + 6z) + (-8y2 + 4y2) Combine (10z) + ( -4y2 ) Remember adding a negative is the same as subtracting 10z – 4y2 Are equal to each other It is best to put in alphabetical order, then descending order of exponent (degree of term) -4y2 + 10z

  24. (12a – b + 6ab) - (11ab – 2ab + 7a) 12a – b + 6ab – 11ab + 2b – 7a Pair up (12a – 7a) + (-b + 2b) + (6ab – 11ab) combine (5a) + (b) + (-5ab) remove 5a + b – 5ab

  25. (2c – 2c2 + c3) – (5c + 3c3 – 2c2) Step one: Step two: 2c – 2c2 + c3 – 5c – 3c3 + 2c2 Step three: (2c – 5c) + ( -2c2 + 2c2 ) + ( c3 – 3c3 ) Step four: (-3c) + (0) + (-2c3 ) Step five: -3c – 2c3 Are equal Step six: -2c3 – 3c

  26. Congrats!!! You have mastered Adding and Subtracting Polynomials!!!!! When you are ready to continued click on the arrow GOOD LUCK!! HAVE FUN!!!

  27. Multiplying exponents is fun!! Before we start multiplying polynomials lets learn about multiplying variables, and how to work with their exponents. *when multiplying a variable you add the exponents together like this: a2 x a2 = a2+2 = a4 b3 x b = = b3+1 = b4 Lets try another We know this is always an exponential value of 1

  28. Next Step: Multiply (monomial)(monomial) *to multiply monomial by monomial you multiply all parts of term 1 by all parts of term 2: which means you multiply bases and exponents of each term (4a2b2) (5a3b5) (20a5b7) Try this example If you are a little overwhelmed don’t worry go to the next slide for a detailed explanation

  29. (4a2b2) (5a3b5) Step 1: multiplying the coefficients 4 x 5 = 20 Step 2: multiply first variable, adding the exponents a2 x a3 a2+3 a5 Step 3: repeat step two for the 2nd variable (continue for as many times as needed b2 x b5 b2+5 b7 Step 4: put them all together 20a5b7

  30. Now try some on your own (2x2 )(6x4) (4ab2)(4a4b) (2xyz)(7x3yz5) (10z)(3xz2) *think carefully* (a2b2c4)(11ab4c7) Answers

  31. Answers When multiplying a variable that does not have an exponent make sure you do not multiply by an exponential value of zero. Even though we don’t write the one it is given that it is there. 2x6 16a5b3 14x4y2z6 30xz3 11a3b6c11 If you had any trouble with these make sure you remember/review the step by step operations

  32. Multiplying: (monomial)(polynomial) (monomial)(polynomial) multiplication is very similar to (monomial)(monomial) multiplication. You must now distribute the monomial more than just one time, you are distributing it to ALL terms in the polynomial. Like so: -2x3(3x2y + 4y3 – 5x4) (-2x3)(3x2y) + (-2x3)(4y3) + (-2x3)(-5x4) -6x5y – 8x3y3 + 10x7 3a2b (a3b – 4a2 + 4bc2) (3a2b)(a3b) + (3a2b)(-4a2) + (3a2b)(4bc2) 3a5b2 – 12a4b + 12a2b2c) 2b(4b2 + 2a – b3) (2b)(4b2) + (2b)(2a) + (2b)(-b3) 8b3 + 4ab – 2b4

  33. Not this type of foil Multiplying: (binomial)(binomial) When multiplying (binomial)(binomial) you need to make sure you multiply both terms of the other polynomial. The easiest method is FOILing F first term O outer term I inner term L last term ( a + b ) ( a + b ) F O I L Remember to combine “like” terms if possible (a)(a) + (a)(b) + (b)(a) + (b)(b) a2 + ab + ba + b2 a2 + 2ab + b2

  34. F O I L (2x + 4y) (8y – 3x) F O I L (2x)(8y) + (2x)(-3x) + (4y)(8y) + (3x)(4y) 16xy – 6x2 + 32y2 + 12xy 28xy – 6x2 + 32y2 (3y + 5y2) (y2 + 5y3) F O I L (3y)(y2) + (3y)(5y3) + (5y2)(y2) + (5y2)(5y3) 3y3 + 15y4 + 5y4 + 25y5 25y5 + 20y4 + 3y3

  35. F O I L If you see this (a +b )2 It does NOT = a2 + b2 (a +b )2 can also be written as (a+b)(a+b) *only if binomials are identical can it be written like (a+b)2 *therefore (a+b)(c+d) can NOT be written in that form * To solve (a + b)2 you FOIL (3x + 4x2)2 (3x + 4x2)(3x + 4x2) (3x)(3x) + (3x)(4x2) + (4x2)(3x) + (4x2)(4x2) 9x2 + 12x3 + 12x3 + 16x4 16x4 + 24x3 + 9x2 (4 + 2y)2 (4 + 2y)(4 + 2y) (4)(4) + (4)(2y) + (2y)(4) + (2y)(2y) 16 + 8y + 8y + 4y2 or

  36. Multiplying: (polynomial)(polynomial) If you said to distribute every term in the first polynomial by every term in the second polynomial you are exactly right Based on your knowledge on multiplying polynomials what would you guess the main idea for (polynomial)(polynomial) would be? *for these you use distributive property twice* (3x + 1) (2x2 + 3x -1 ) (3x)(2x2) + (3x)(3x) + (3x)(-1) + (1)(2x2) + (1)(3x) + (1)(-1) 6x3 + 9x2 -3x + 2x2 + 3x -1 6x3 + (9x2 + 2x2) + (-3x + 3x) – 1 6x3 + 11x2 -1 *combine like terms

  37. Multiplying: (polynomial)(polynomial) *take your time and check over your answer (-2x – y2 )(10y2 – 2xy + 7y) (-2x)(10y2) + (-2x)(-2xy) + (-2x)(7y) + (-y2)(10y2) + (-y2)(-2xy) + (-y2)(7y) -20xy2 + 4x2y -14xy – 10y4 + 2xy3 – 7y3 (3a + 3ab2)(a2 + 4b + 8) (3a)(a2) + (3a)(4b) + (3a)(8) + (3ab2)(a2) + (3ab2)(4b) + (3ab2)(8) 3a3 + 12ab + 24a + 3a3b2 + 12ab3 + 24ab2 *distribute twice Try to combine like terms if possible *distribute twice

  38. Let’s Check Your Knowledge

  39. Knowledge Check Solve: (4a2bc4) (10a4bc3) 40a8bc7 40a6b2c7 40a6bc12 40abc Solution

  40. Congrats!!! You got it!! You really know your stuff

  41. Oops!!! That’s not it Try again

  42. (4a2 bc4)(10a4bc3) • We have (monomial)(monomial) • Multiply coefficients 40 • Now multiply each variable 40a2+4 b1+1 c4+3 • by adding the exponents • 40a6b2c7

  43. Knowledge Check Solve: (2 – 9a)2 4 – 9a2 4 + 36a + 81a2 4 – 81a2 4 – 36a + 81a2 Solution

  44. (2-9a)2 Did you remember to write it as (binomial)(binomial) (2-9a)(2-9a) FOIL F O I L (2)(2) + (2)(-9a) + (-9a)(2) + (-9a)(-9a) 4 – 18a -18a + 81a2 Combine like terms 4 – 36a + 81a2

  45. Knowledge Check Solve: 8y(4y – 6x2y + 7y2) 32y2 – 48x2y2 + 56y3 32y + 48x2y2 + 56y3 32y – 48y2 + 56y3 32y2 + 48y + 56y2 solution

  46. 8y(4y – 6x2y + 7y2) Distribute 8y to all terms (8y)(4y) + (8y)(-6x2y) + (8y)(7y2) Remember even though 8y does not have an x to multiply it is as if multiplying by 1…not 0 32y2 - 48x2y2 + 56y3

  47. Knowledge Check Solve: (a2 + b2 + c2) (a2 + 2bc) a4 + a2b2 + a2c2 + 2a2bc + 2b3c + 2bc3 a4 + 2a2b2 + a2c2 + 2a2bc + 2b3c + 2bc2 a2 + a2b2 + a2c2 + 2a2bc + 2b3c + 2bc3 a + ab + a2c2 + a2bc + 2b3c + 2bc3 Solution

  48. (a2 + b2 + c2) (a2 + 2bc) Be extremely careful because all the exponents and different variables can get confusing (a2)(a2) + (a2)(b2) + (a2)(c2) + (2bc)(a2) + (2bc)(b2) + (2bc)(c2) a4 + a2b2 + a2c2 + 2a2bc + 2b3c + 2bc3

  49. Knowledge Check Solve: (2a7b2cd2)(4c2d2) 8c3d4 8a7b2c3d4 8abc2d4 8abcd Solution

  50. (2a7b2cd2)(4c2d2) • Multiply coefficients 8 • Variable by adding exponents 8a7b2c3d4 • Reminder multiplying a7b2 is as multiplying by 1

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