html5-img
1 / 35

10.1 Adding and Subtracting Polynomials

The degree is 3. The leading coefficient is 6. The constant term is 15. 10.1 Adding and Subtracting Polynomials. A polynomial of two terms is a binomial . 7 xy 2 + 2 y. 8 x 2 + 12 xy + 2 y 2. A polynomial of three terms is a trinomial .

wynn
Download Presentation

10.1 Adding and Subtracting Polynomials

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. The degree is 3. The leading coefficient is 6. The constant term is 15. 10.1 Adding and Subtracting Polynomials A polynomial of two terms is a binomial. 7xy2 + 2y 8x2 + 12xy + 2y2 A polynomial of three terms is a trinomial. The leading coefficient of a polynomial is the coefficient of the variable with the largest exponent. The constant term is the term without a variable. 6x3 – 2x2 + 8x + 15

  2. Equation Degree Function linear f(x) = mx + b one quadratic f(x) = ax2 + bx + c, a 0 two cubic f(x) = ax3 + bx2 + cx + d, a 0 three 10.1 Adding and Subtracting Polynomials The degree of a polynomial is the greatest of the degrees of any of its terms. The degree of a term is the sum of the exponents of the variables. Examples: 3y2 + 5x +7 degree 2 21x5y + 3x3+2y2 degree 6 Common polynomial functions are named according to their degree.

  3. 10.1 Adding and Subtracting Polynomials To add polynomials, combine like terms. Examples: Add (5x3 + 6x2 + 3) + (3x3 – 12x2 – 10). Use a horizontal format. Rearrange and group like terms. (5x3 + 6x2 + 3) + (3x3 – 12x2 – 10) = (5x3 + 3x3 ) + (6x2 – 12x2) + (3 – 10) = 8x3 – 6x2 – 7 Combine like terms.

  4. 10.1 Adding and Subtracting Polynomials Add (6x3 + 11x –21) + (2x3 + 10 – 3x) + (5x3 + x – 7x2 + 5). Use a vertical format. 6x3 + 11x – 21 Arrange terms of each polynomial in descending order with like terms in the same column. 2x3 – 3x + 10 5x3 – 7x2 + x + 5 13x3 – 7x2 + 9x – 6 Add the terms of each column.

  5. 10.1 Adding and Subtracting Polynomials The additive inverse of the polynomial x2 + 3x + 2 is –(x2 + 3x + 2). This is equivalent to the additive inverse of each of the terms. –(x2 + 3x + 2) = – x2 – 3x – 2 To subtract two polynomials, add the additive inverse of the second polynomial to the first.

  6. 10.1 Adding and Subtracting Polynomials Example: Add (4x2 – 5xy + 2y2) – (–x2 + 2xy – y2). (4x2 – 5xy + 2y2) – (–x2 + 2xy – y2) Rewrite the subtraction as theaddition of the additive inverse. = (4x2 – 5xy + 2y2) + (x2 – 2xy + y2) = (4x2 + x2) + (–5xy – 2xy) + (2y2 + y2) Rearrange and group like terms. = 5x2 – 7xy + 3y2 Combine like terms.

  7. 10.1 Adding and Subtracting Polynomials Let P(x) = 2x2 – 3x + 1 and R(x) = – x3 + x + 5. Examples: Find P(x) + R(x). P(x) + R(x) = (2x2 – 3x + 1) + (– x3 + x +5) = – x3 + 2x2 + (–3x + x) + (1 + 5) = – x3 + 2x2 – 2x + 6

  8. 10.2 Multiplying Polynomials To multiply a polynomial by a monomial, use the distributive property and the rule for multiplying exponential expressions. Examples:. Multiply:2x(3x2 + 2x – 1). = 2x(3x2) + 2x(2x)+ 2x(–1) = 6x3 + 4x2 – 2x

  9. 10.2 Multiplying Polynomials Multiply:–3x2y(5x2 – 2xy + 7y2). = –3x2y(5x2) – 3x2y(–2xy) – 3x2y(7y2) = –15x4y + 6x3y2 – 21x2y3

  10. 10.2 Multiplying Polynomials To multiply two polynomials, apply the distributive property. Example: Multiply: (x – 1)(2x2 + 7x + 3). = (x – 1)(2x2) + (x – 1)(7x) + (x – 1)(3) = 2x3 – 2x2 + 7x2 – 7x + 3x – 3 = 2x3 + 5x2 – 4x – 3

  11. Example: 2x2 + 7x + 3 x – 1 10.2 Multiplying Polynomials Example: Multiply: (x – 1)(2x2 + 7x + 3). Two polynomials can also be multiplied using a vertical format. – 2x2 – 7x – 3 Multiply –1(2x2 + 7x + 3). 2x3 + 7x2 + 3x Multiply x(2x2 + 7x + 3). Add the terms in each column. 2x3 + 5x2 – 4x – 3

  12. 10.2 Multiplying Polynomials To multiply two binomials use a method called FOIL, which is based on the distributive property. The letters of FOIL stand for First, Outer, Inner, and Last. 1. Multiply the first terms. 4. Multiply the last terms. 2. Multiply the outer terms. 5. Add the products. 3. Multiply the inner terms. 6. Combine like terms.

  13. Last Outer Inner First 10.2 Multiplying Polynomials Examples: Multiply:(2x +1)(7x – 5). = 2x(7x) + 2x(–5) + (1)(7x) + (1)(–5) = 14x2 – 10x + 7x – 5 = 14x2 – 3x – 5

  14. First Last Outer Inner 10.2 Multiplying Polynomials Multiply:(5x – 3y)(7x + 6y). = 5x(7x) + 5x(6y) + (– 3y)(7x) + (– 3y)(6y) = 35x2 + 30xy – 21yx – 18y2 = 35x2 + 9xy – 18y2

  15. 10.3 Special Cases The multiply the sum and difference of two terms, use this pattern: (a + b)(a – b) = a2 – ab + ab – b2 = a2 – b2 square of the second term square of the first term

  16. 10.3 Special Cases Examples: (3x + 2)(3x – 2) (x + 1)(x – 1) = (3x)2– (2)2 = (x)2– (1)2 = 9x2– 4 = x2– 1

  17. 10.3 Special Cases To square a binomial, use this pattern: (a + b)2 = (a + b)(a + b) = a2 + ab + ab + b2 = a2 + 2ab + b2 square of the first term twice the product of the two terms square of the last term

  18. 10.3 Special Cases Examples: Multiply: (2x – 2)2 . = (2x)2 + 2(2x)(– 2) +(– 2)2 = 4x2 – 8x + 4 Multiply: (x + 3y)2 . = (x)2 + 2(x)(3y) +(3y)2 = x2 + 6xy + 9y2

  19. 10.4 Factoring The simplest method of factoring a polynomial is to factor out the greatestcommon factor (GCF) of each term. Example: Factor 18x3+ 60x. Find the GCF. GCF = 6x 18x3+ 60x=6x(3x2) + 6x(10) Apply the distributive law to factor the polynomial. = 6x(3x2 + 10) Check the answer by multiplication. 6x(3x2 + 10) = 6x(3x2) + 6x(10) = 18x3+ 60x

  20. 10.4 Factoring Example: Factor 4x2 – 12x + 20. Therefore, GCF = 4. 4x2 – 12x + 20 = 4x2 – 4 · 3x + 4 · 5 = 4(x2 – 3x + 5) Check the answer. 4(x2 – 3x + 5) = 4x2 – 12x + 20

  21. 10.4 Factoring A common binomial factor can be factored out of certain expressions. Example: Factor the expression 5(x + 1) – y(x + 1). Check. 5(x + 1) – y(x + 1) = (5 – y)(x + 1) (5 – y)(x + 1)= 5(x + 1) – y(x + 1)

  22. 10.4 Factoring A difference of squares can be factored using the formula a2 – b2 = (a + b)(a – b). Example: Factor x2 – 9y2. = (x)2– (3y)2 x2 – 9y2 Write terms as perfect squares. = (x + 3y)(x– 3y) Use the formula.

  23. 10.4 Factoring The same method can be used to factor any expression which can be written as a difference of squares. Example: Factor 4(x + 1)2 – 25y4. = (2(x + 1))2 – (5y2)2 4(x + 1)2 – 25y4 = [(2(x + 1)) + (5y2)][(2(x + 1)) – (5y2)] = (2x + 2 + 5y2)(2x + 2 – 5y2)

  24. 10.4 Factoring Some polynomials can be factored by grouping termsto produce a common binomial factor. Examples: Factor 2xy + 3y – 4x – 6. 2xy + 3y – 4x – 6 = (2xy + 3y) – (4x + 6) Group terms. = (2x + 3)y – (2x + 3)2 Factor each pair of terms. = (2x + 3)(y – 2) Factor out the common binomial.

  25. 10.4 Factoring 2a2 +3bc– 2ab –3ac Factor 2a2 +3bc– 2ab –3ac. =2a2– 2ab+3bc–3ac Rearrange terms. = (2a2– 2ab)+(3bc–3ac) Group terms. = 2a(a– b)+3c(b –a) Factor. = 2a(a– b)–3c(a –b) b – a = –(a – b). = (2a–3c)(a –b) Factor.

  26. 10.4 Factoring To factor a trinomial of the form x2 + bx + c, express the trinomial as the product of two binomials. For example, x2 + 10x + 24 = (x +4)(x +6).

  27. 10.4 Factoring One method of factoring trinomialsis based on reversing the FOIL process. Express the trinomial as a product of two binomials with leading term x and unknown constant terms a and b. Example: Factor x2 + 3x + 2. x2 + 3x + 2 = (x + a)(x + b) F O I L = x2 + ax + bx + ab Apply FOIL to multiply the binomials. Since ab = 2 and a + b = 3, it follows that a = 1 and b = 2. = x2 + (a + b)x + ab = x2 + (1 +2)x + 1 · 2 Therefore, x2 + 3x + 2 = (x + 1)(x + 2).

  28. 10.4 Factoring = (x+ a)(x+b) Example: Factor x2 – 8x + 15. = x2+ (a + b)x + ab and ab = 15. Therefore a + b = –8 It follows that both a and b are negative. x2 – 8x + 15 = (x – 3)(x – 5). Check: (x – 3)(x – 5) = x2– 5x– 3x + 15 = x2 – 8x + 15.

  29. 10.4 Factoring Example: Factor x2 + 13x + 36. = (x+ a)(x+b) = x2+ (a + b)x + ab Therefore a and b are two positive factors of 36 whose sum is 13. = (x + 4)(x+ 9) x2 + 13x + 36 Check: (x + 4)(x + 9) = x2 + 9x + 4x + 36 = x2 + 13x + 36.

  30. 10.4 Factoring Example: Factor 4x3 – 40x2 + 100x. 4x3 – 40x2 + 100x = 4x(x2) – 4x(10x) + 4x(25) The GCF is 4x. = 4x(x2 – 10x + 25) Use distributive property to factor out the GCF. = 4x(x – 5)(x – 5) Factor the trinomial. Check: 4x(x – 5)(x – 5) = 4x(x2 – 5x – 5x + 25) = 4x(x2 – 10x + 25) = 4x3 – 40x2 + 100x

  31. 10.4 Factoring Example: Factor 2x2+5x +3. 2x2+5x +3 = (2x + a)(x + b) For some a and b. = 2x2 + (a + 2b)x + ab 2x2+5x +3 = (2x + 3)(x + 1) Check: (2x + 3)(x + 1) = 2x2+2x+3x+3 = 2x2+5x +3.

  32. 10.4 Factoring Example: Factor 4x2 – 12x + 5. Since ab= +5, aandbhave the same sign. This polynomial factors as (x + a)(4x + b) or (2x + a)(2x + b). a = –1, b = –5 or a = 1 and b = 5. The middle term –12xequals either (4a + b)x or (2a + 2b)x. Since a and b cannot both be positive, they must both be negative. 4x2 – 12x + 5= (2x –1)(2x – 5)

  33. Trinomials which are quadratic in form are factored like quadratic trinomials. Example: Factor 3x4 + 28x2 + 9. 3x4 + 28x2 + 9 = 3u2 + 28u + 9 Let u = x2. = (3u +1)(u + 9) Factor. = (3x2 + 1)(x2 + 9) Replace u by x2. Many trinomials cannot be factored. Example: Factorx2 + 3x + 5. Let x2 + 3x + 5 = (x + a)(x + b) = x2 + (a + b)x + ab. Then a + b= 3 and ab=5. This is impossible. The trinomial x2 + 3x + 5 cannot be factored.

  34. Factor by GroupingExample 2: • FACTOR: 6mx – 4m + 3rx – 2r • Factor the first two terms: 6mx – 4m= 2m(3x - 2) • Factor the last two terms: + 3rx – 2r = r(3x - 2) • The green parentheses are the same so it’s the common factor Now you have a common factor (3x - 2) (2m + r)

  35. x – 6 x + 5 Example: The length of a rectangle is (x + 5) ft. The width is (x – 6) ft. Find the area of the rectangle in terms of the variable x. A = L · W = Area L = (x + 5) ft W = (x – 6) ft A = (x + 5)(x – 6 ) = x2 – 6x + 5x – 30 = x2 – x – 30 The area is (x2 – x – 30) ft2.

More Related