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Polynomials

Polynomials. Polynomial – An algebraic expression that contains only positive integral exponent and containing no variable in the denominator. Examples:. Polynomial or Not. 3x 2 – 2x + 5. Polynomial or Not. 4x -3 – 2x + 10. Polynomial or not. Important terms

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Polynomials

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  1. Polynomials

  2. Polynomial – An algebraic expression that contains only positive integral exponent and containing no variable in the denominator.

  3. Examples:

  4. Polynomial or Not 3x2 – 2x + 5

  5. Polynomial or Not 4x-3 – 2x + 10

  6. Polynomial or not

  7. Important terms Degree – the sum of the exponents of the variables and it must be the highest degree in that polynomial. Constant – a symbol that never changes its value. Variable – a letter that represents a number. Term – it is either a constant or product of a constant and a variable connected by a plus or minus signs. Literal Coefficient – refers to the variables of a term. Numerical Coefficient – the number or constant in a term.

  8. How many terms are there? 3x - 5

  9. How many terms are there? 4x2 – 3x - 10

  10. How many terms are there? a – b + c - d

  11. What is the degree? 3x3 + 4x2 + 9x - 10

  12. What is the degree? 5a2bc

  13. Classification of Polynomials according to the number of terms. • 1 Monomial – A polynomial of one term (Ex. 2xy) • 2. Binomial – A polynomial of two terms. (Ex. 2xy – 3y) • 3. Trinomial – A polynomial of three terms ( Ex. 2xy – 3y + 5x2) • 4. Multinomial – A polynomial with four or more term . • ` ( Ex. 2a-3b+4c+d)

  14. Classification of Polynomials according to degree. • Linear – first degree polynomial Example: x + 5 2. Quadratic – second degree polynomial Example: x2 + x + 5 3. Cubic – third degree polynomial Example: x3 + x2 + x + 5 4. Quartic – fourth degree polynomial Example: x4 + x3 + x2 + x + 5 5 Quintic – fifth degree polynomial Example: x5 + x4 + x3 + x2 + x + 5

  15. Complete the table.

  16. Topic: Addition and Subtraction of Polynomials • Adding polynomials is just a matter of combining like terms, with some order of operations. • Addition or Subtraction may be done either horizontally or vertically depending on how the problem is presented.

  17. Examples: 1. Add 3x2 + 5x + 3 and x2 + x – 2 Solution: (3x2 + x2) + (5x+x) + (3 + (-2) ) 4x2 + 6x + 1 2. Add 2x2 + 5x – 3 and 10x2 – 3x – 2 Solution: (2x2+10x2) + (5x+(-3x) ) + (-3 + (-2) ) 12x2+ 2x - 5

  18. Subtraction of Polynomials • Subtracting polynomials is quite similar to adding polynomials. To subtract one polynomial from another polynomial, simply change the sign of each term of the polynomial in the subtrahend, then proceed to addition of polynomials.

  19. Examples: 1.Subtract: (6x2 – 2x + 8) – (4x2 – 11x + 10) Solution: (6x2 – 2x + 8) + (-4x2 + 11x – 10) (6x2 + (-4x2) ) + (-2x + 11x) + (8 + (-10) ) 2x2 + 9x – 2 2. Subtract: (12x2 + 13x + 16) – (11x2 + 16x + 8 Solution: (12x2 + 13x + 16 ) + (-11x2 – 16x – 8) • (12x2 +(-11x2) ) + (13x + (-16x) ) + (16 + (-8) ) • x2 – 3x + 8

  20. Board Drill (5x + 8) + (2x + 11)

  21. Board Drill (3x – 6) + (-6x + 1)

  22. Board Drill (4x2 – 5x + 7) + (5x2 – 10x + 9)

  23. Board Drill (3x2 – 7x + 8) + (4x2 + 5x – 1)

  24. Board Drill (7x + 9) - (2x + 5)

  25. Board Drill (5x – 7) – (4x + 9)

  26. Board Drill (4x2 + 5x + 6) – (2x2 + 3x + 1)

  27. Board Drill (5x2 – 7x + 4) – (3x2 + 7x – 6)

  28. Seatwork: Perform the indicated operation. • (8x + 6) + (9x – 3) • (3x2 – 5x + 2) + (4x2 – 3x + 8) • (7x2 + 10x – 1) + (-x2 + 3x + 4) • (4x2 + 5x + 9 ) – (3x2 + 4x + 1) • (2x2 – 8x + 1) – (4x2 + 6 + 1)

  29. Laws of Exponents • Recall that in xm, x is the base and m is the exponent. The exponent tells the number of times the base x is takenas a factor. To multiply and divide polynomials, you have to be equippedwith the knowledge of the laws of exponents. The laws of exponents will be your guide in multiplying and dividing polynomials.

  30. Law 1: Product of a Power Rule • To multiply powers of the same base, copy the base and add the exponents. xm* xn = x m+n Examples: • x4 * x3 = x7 • x5 * x8 = x13 • 4x2 * 3x = 12x 3

  31. Law 2: Power of a Power Rule • To find a power of a product, determine the powerof each factor and then multiply. • (xm)n = xmn Examples: • (x3)2 = x6 • (x4)5 = x20 • (2x4)2 = 4x8

  32. Law 3: Power of a Product Rule • To find a power of a product, determine the powerof each factor and then multiply. • (xy)n = xnyn • Examples: 1. (x2y3)4 = x8y 12 2. (x5y2)3 = x15y6 3. (3x2y)2 = 9x4y2

  33. Laws 4: Quotient of a Power • For every integer m and n, and x ≠ 0 • = x m-n if m > n • = if m < n Examples: 1. = x 2. =

  34. Board Drill x3 * x9

  35. Board Drill (x4)6

  36. Board Drill (x3y4)7

  37. Board Drill

  38. Board Drill (3x2)2

  39. Multiplication of Polynomials • In multiplying a polynomial by another polynomial using the distributive property and using the different methods, multiply each of the multiplicand by every term of the multiplier, then add. • Foil Method is only used if there are two terms involved in both the multiplicand and the multiplier.

  40. Examples: Multiply the following polynomials 1. 2x ( x2 + 2x ) = 2x3 + 4x2 2. x3 ( x2 + 2x – 1 ) = x5 + 2x4 – x3 3. (4x + 5) (x-2) = 4x2 – 3x – 10 4. (x+3) (2x+ 3) = 2x2 + 9x +9

  41. Division of Polynomials • To divide a polynomial of two or more terms by a monomial, divide each term of the dividend by the divisor and simplify. Examples: Divide the following polynomials. • 1. = 5a + 4b • 2. = 9x – 4y • 3. = 4ax – 2b • 4. = mn -1

  42. Seatwork Perform the indicated operation. • x3 (x2 + 2x – 3) • 2a(3a2 + 4a – 8) • (x+4) (x+3) • 12ax – 9b 3

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