Description NAME DATE PERIOD 55 Study Guide and Intervention Solving Polynomial Equations Factor Polynomials For any number of terms, check for: greatest common factor For two terms, check for: Difference of Fill & Sign Online, Print, Email, Fax, or Download
Not the form you were looking for? Сomplete the NAME DATE PERIOD 55 for freeIf you believe that this page should be taken down, please follow our DMCA take down process here. NAME 5-5 DATE PERIOD Study Guide and Intervention Solving Polynomial Equations Factor Polynomials For any number of terms, check for: greatest common factor For two terms, check for: Difference of two squares a 2 - b 2 = (a + b)(a - b) Sum of two cubes a 3 + b 3 = (a + b)(a 2 - ab + b 2) Difference of two cubes a 3 - b 3 = (a - b)(a 2 + ab + b 2) Techniques for Factoring Polynomials For three terms, check for: Perfect square trinomials a 2 + 2ab + b 2 = (a + b)2 a 2 - 2ab + b 2 = (a - b)2 General trinomials acx 2 + (ad + bc)x + bd = (ax + b)(cx + d) For four or more terms, check for: Grouping ax + bx + ay + by = x(a + b) + y(a + b) = (a + b)(x + y) Factor 24x2 - 42x - 45. First factor out the GCF to get 24x2 - 42x - 45 = 3(8x2 - 14x - 15). To find the coefficients of the x terms, you must find two numbers whose product is 8 · (-15) = -120 and whose sum is -14. The two coefficients must be -20 and 6. Rewrite the expression using -20x and 6x and factor by grouping. 8x2 - 14x - 15 = 8x2 - 20x + 6x - 15 Group to find a GCF. = 4x(2x - 5) + 3(2x - 5) Factor the GCF of each binomial. = (4x + 3)(2x - 5) Distributive Property 2 Thus, 24x - 42x - 45 = 3(4x + 3)(2x - 5). Exercises Factor completely. If the polynomial is not factorable, write prime. 1. 14x2y2 + 42xy3 14xy 2(x + 3y) 4. x4 - 1 (x 2 + 1)(x + 1)(x - 1) 7. 100m8 - 9 (10m 4 - 3)(10m 4 + 3) Chapter 5 2. 6mn + 18m - n - 3 (6m - 1)(n + 3) 5. 35x3y4 - 60x4y 5x 3y(7y 3 - 12x) 8. x2 + x + 1 prime 3. 2x2 + 18x + 16 2(x + 8)(x + 1) 6. 2r3 + 250 2(r + 5)(r 2 - 5r + 25) 9. c4 + c3 - c2 - c c(c + 1)2(c - 1) 29 Glencoe Algebra 2 Lesson 5-5 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Example NAME DATE 5-5 Study Guide and Intervention PERIOD (continued) Solving Polynomial Equations Solve Polynomial Equations If a polynomial expression can be written in quadratic form, then you can use what you know about solving quadratic equations to solve the related polynomial equation. Example 1 Solve x 4 - 40x 2 + 144 = 0. x4 - 40x2 + 144 = 0 Original equation 2 2 2 (x ) - 40(x ) + 144 = 0 Write the expression on the left in quadratic form. 2 2 (x - 4)(x - 36) = 0 Factor. 2 x -4=0 or x2 - 36 = 0 Zero Product Property (x - 2)(x + 2) = 0 or (x - 6)(x + 6) = 0 Factor. x - 2 = 0 or x + 2 = 0 or x - 6 = 0 or x + 6 = 0 Zero Product Property x = 2 or x = -2 or x = 6 or x = -6 Simplify. The solutions are ±2 and ±6. Example 2 x - 15 = 0. Solve 2x + √ 2x + √x - 15 = 0 2(√x ) 2+ √x - 15 = 0 (2 √x -5)(√x + 3) = 0 2 √x - 5 = 0 or √x + 3 = 0 √x = -3 Write the expression on the left in quadratic form. Factor. Zero Product Property Simplify. Since the principal square root of a number cannot be negative, √x = -3 has no solution. 25 1 The solution is − or 6 − . 4 4 Exercises Solve each equation. 1. x4 = 49 , ±i √ ± √7 7 4. 3t6 - 48t2 = 0 0, ±2, ±2i 7. x4 - 29x2 + 100 = 0 ±5, ±2 10. x - 5 √x + 6 = 0 4, 9 Chapter 5 2. x4 - 6x2 = -8 3. x4 - 3x2 = 54 ±2, ± √ 2 ±3, ±i √ 6 5. m6 - 16m3 + 64 = 0 2, -1 ± i √ 3 8. 4x4 - 73x2 + 144 = 0 3 ±4, ± − 6. y4 - 5y2 + 4 = 0 ±1, ±2 7 1 9. − -− 2 x + 12= 0 x 1 1 − ,− 2 3 4 11. x - 10 √x + 21 = 0 9, 49 2 − 1 − 12. x 3 - 5x 3 + 6 = 0 27, 8 30 Glencoe Algebra 2 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 5 √x or =− 2 Original equation This preview shows page 1 - 2 out of 4 pages. NAME ______________________________________________ DATE______________________________ PERIOD ______________5-5 Study Guide and InterventionSolving Polynomial EquationsFactor PolynomialsTechniques for FactoringPolynomialsFor any number of terms, check for:greatest common factorFor two terms, check for:Difference of two squaresa2–b2= (a+b)(a–b)Sum of two cubesa3+b3= (a+b)(a2–ab+b2)Difference of two cubesa3–b3= (a–b)(a2+ab+b2)For three terms, check for:Perfect square trinomialsa2+ 2ab+b2=(a+b)2a2– 2ab+b2=(a−b)2General trinomialsacx2+ (ad+bc)x+bd= (ax+b)(cx+d)For four or more terms, check for:Groupingax+bx+ay+by=x(a+b) +y(a+b)= (a+b)(x+y)Example:Factor24x2– 42x– 45.First factor out the GCF to get24x2– 42x– 45 = 3(8x2– 14x– 15). To find the coefficients of thexterms, youmust find two numbers whose product is 8⋅(–15) = –120 and whose sum is –14. The two coefficients must be –20 and 6.Rewrite the expression using –20xand 6xand factor by grouping.8x2– 14x– 15 =8x2– 20x+ 6x– 15Group to find a GCF. Upload your study docs or become a Course Hero member to access this document Upload your study docs or become a Course Hero member to access this document End of preview. Want to read all 4 pages? Upload your study docs or become a Course Hero member to access this document Professor paola dolcemascolo Tags Elementary algebra, 100 m, 16 m, 4 M, 18m |