Polynomial Functions - NJCTL

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Polynomial FunctionsNOTE: Some problems in this file are used with permission from the website of the New York State Department of Education. Various files. Internet. Available from . Accessed August, 2014. Properties of Exponents: Class WorkSimplify the following expressions.-4g3h2j-2-34k33mn223p7q32p2q23-25r3s4t22r3s-343u2v-436u4v3-28w2x-3y4z512w3x-4y5z-6-3Properties of Exponents: HomeworkSimplify the following expressions.-3g-4h3j-3-44k46m3n-428p7q92p2q24-245r10s12t82r4s-5-36u6v-339u5v-6-26w-3x-4y5z615w3x-4y5z-6-2Operations with Polynomials: Class WorkDetermine if each function is a polynomial function. If so, write it in standard form, name its degree, state its type based on degree and based on number of terms, and identify the leading coefficient.2x2+3x247y-3y2+3y5a3-2a-4a+36a2b-5ab2+2ab22x-2-4+-5x-2-3Perform the indicated operations. 4g2-2-3g+5+2g2-g6t-3t2+4-(t2+5t-9)7x5+8x4-3x+(5x4+2x3+9x-1)-10x3+4x2-5x+9-(2x3-2x2+x+12)The legs of an isosceles triangle are (3x2+ 4x +2) inches and the base is (4x-5) inches. Find the perimeter of the triangle.-2a4a2b-3ab2-6ab7jk25j3k+9j2-2k+102x-34x+2c2-3c+4m-32m2+4m-52f+56f2-4f+13t2-2t+94t2-t+1The width of a rectangle is (5x+2) inches and the length is (6x-7) inches. Find the area of the rectangle.The radius of the base of a cylinder is (3x + 4) cm and the height is (7x + 2) cm. Find the volume of the cylinder (V = πr2h). A rectangle of (2x) ft by (3x-1) ft is cut out of a large rectangle of (4x+1)ft by (2x+2)ft. What is area of the shape that remains?A pool that is 20ft by 30ft is going to have a deck of width x ft added all the way around the pool. Write an expression in simplified form for the area of the deck.Multiply and simplify:b+22c-1c-12d+4e25f+95f-9What is the area of a square with sides (3x+2) inches? Expand, using the Binomial Theorem:2x+4y57a+b33x-4z6y-5z4Operations with Polynomials: HomeworkDetermine if each function is a polynomial function. If so, write it in standard form, name its degree, state its type based on degree and based on number of terms, and identify the leading coefficient.2x2+0.4x347y-8y2+9y11a4-2a3+7a2-8a+96a211-5a9+22x23-4+-5x2-3 Perform the indicated operations:3n-13-2n2+4n-6-(5n-4)5g2-4-3g3+7+5g2-5g-8x4+7x3-3x+5+(5x4+2x2-16x-21)17x3-9x2+5x-18-(11x3-2x2-19x+15)The width of a rectangle is (5x2+6x +2) inches and the length is (6x-7) inches. Find the perimeter of the rectangle.4x3x2-5x-2-6a3a2b-5ab2-7b8j2k32j3k+6j2-5k+114x+56x+12b-94b-22c2-43c+22m-53m2-6m-43f+46f2-4f+12p2-5p2+8p+25t2-3t+63t2-2t+1The width of a rectangle is (4x-3) inches and the length is (3x-5) inches. Find the area of the rectangle.The radius of the base of a cone is (9x - 3) cm and the height is (3x + 2) cm. Find the volume of the cylinder (V = 13πr2h). A rectangle of (3x) ft by (5x-1) ft is cut out of a large rectangle of (6x+2)ft by (3x+4)ft. What is area of the shape that remains?A pool that is 25ft by 40ft is going to have a deck of width (x + 2) ft added all the way around the pool. Write an expression in simplified form for the area of the deck.Multiply and simplify:3a-13a+1b-22c-1c+13d-5e25f+95f+9What is the area of a square with sides of (4x-6y) inches? Expand the following using the binomial Theorem:2a-b63x+2y35y-4z5a+7b4Factoring I ClassworkFactoring out the GCF 6x3y2 – 3x2y 10p3q – 15p3q2 – 5p2q2 7m3n3 – 7m3n2 + 14m3Factoring ax2 + bx + cx2 – 5x – 24m2 – mn – 6n2x2 – 2xy + y2a2 + ab – 12b2x2 – 6xy + 8y22x2 + 7x + 36x2 – x – 25a2 + 17a – 126m2 - 5mn + n26p2 + 37p + 64c2 + 20cd + 25d2Factoring I HomeworkFactoring out the GCF 8x3y – 4x2y2 8m3n3 – 4m2n3 – 32mn3-18p3q2 + 3pqFactoring ax2 + bx + cm2 – 2m – 24a2 – 13a + 12n2 + n – 6x2 – 10xy + 21y2x2 + 11xy + 18y26x2 – 5x + 115p2 – 22p – 510m2 + 13m – 312x2 – 7xy + y24p2 + 24p + 35 15m2 – 13mn + 2n2Spiral Review105. Simplify:106. Multiply:107. Divide108. Evaluate, use x = 5:5 – 4 [(-2) – (-2)] 234 ?423 234 ÷423-2(-6x – 9) + 4Factoring II Classwork Factoring a2 – b2, a3 – b3, a3 + b3a3 – 125x2 – 16y2121a2 – 16b227x3 + 8y3a3b3 – c34x2y2 – 1Factoring by Grouping2xy + 5x + 8y + 209mn – 3m – 15n + 52xy – 10x – 3y + 1510rs – 25r + 6s – 1510pq – 2p – 5q + 110mn + 5m + 6n + 3Mixed Factoring3x3 – 12x2 + 36x6m3 + 4m2 – 2m3a3b – 48ab54x4 + 2xy3x4y + 12x3y + 20x2yFactoring II HomeworkFactoring a2 – b2, a3 – b3, a3 + b3y3 + 2764m3 – 1p2 – 36q2m2n2 – 4x2 + 168x3 – 27y3Factoring by Grouping6mp – 2m – 15p + 56xy + 15x + 4y + 104rs – 4r + 3s – 36tr – 9t – 2r + 38mn + 4m + 6n + 33xy – 4x – 15y + 20Mixed Factoring3m3 – 3mn2-6x3 – 28x2 + 10x18a3b – 50abx4y + 27xy-12r3 – 21r2 – 9r2x2y2 – 2x2y – 2xy2 + 2xySpiral Review144. Simplify:145. Simplify:146. Add:147. Evaluate, use x = -3, y = 28(-4) (2)(-1) + (4)2 172 - (12 - 4)2 + 2 227+535-3x + 2y – xy + x Division of Polynomials: Class WorkSimplify.6x3-3x2+9x3x4a4b3+8a3b3-6a2b2÷2a2b6x3-4x2+7x+33x+14a4+8a3-6a2+3a+4÷a-1Consider the polynomial function fx=3x2+8x-4. Divide f by x-2.b. Find f(2).Consider the polynomial function gx=x3-3x2+6x+8. Divide g by x+1.Find g(-1).Consider the polynomial Px=x3+x2-10x-10.Is x+1 one of the factors of P? Explain.The volume a hexagonal prism is 3t3-4t2+t+2 cm3 and its height is (t+1) cm. Find the area of the base. (Use V=Bh)Division of Polynomials: HomeworkSimplify.16x5-12x3+24x24x24a4b3+8a3b3-16a2b2÷4ab23f3+18f-123f2-13x3-3x2+9x+2x+3Consider the polynomial function fx=x3-24. Divide f by x-2.b. Find f(2).Consider the polynomial function gx=x3+5x2-8x+7. Divide g by x+1.Find g(-1).Consider the polynomial Px=2x3+5x2-12x+5.Is x-1 one of the factors of P? Explain.8f32f+4-1The volume a hexagonal prism is 4t3-3t2+2t+2 cm3 . The area of the base, B is (t-1) cm2. Find the height of the prism. (Use V=Bh)Consider the polynomial Px=x4+3x3-28x2-36x+144.Is 1 a zero of the polynomial P?Is x+3 one of the factors of P?Characteristics of Polynomial Functions: Class WorkFor each function or graph answer the following questions:Does the function have even degree or odd degree?Is the lead coefficient positive or negative?Is the function even, odd or neither?335280061595002755902032000166. 167.307340026034004889502603500168. 169.Is each function below odd, even or neither? fx=2x4+3x2-2y=5x5-3x+1gx=-2x4x2-3xhx=4xFor each function in #’s 170 – 173 above, describe the end behavior in these terms: as x?∞, f(x) ?____, and as x ? -∞, f(x) ?_____. Is each function below odd, even or neither? How many zeros does each function appear to have? 4819650209550032829502095500181610020955002755906286500175. 176.177. 178.Characteristics of Polynomial Functions: HomeworkFor each function or graph answer the following questions:Does the function have even degree or odd degree?Is the lead coefficient positive or negative?Is the function even, odd or neither?307340034290002755905588000179. 180.3143250139700006286506921500181. 182.Is each function below an odd-function, an even-function or neither.183. fx=5x4-6x2+3x184. y=5x5-3x3+1x185. gx=2x24x3-3x186. hx=-45x2+2187. For each function in #’s 183 – 186 above, describe the end behavior in these terms: as x?∞, f(x) ?____, and as x ? -∞, f(x) ?_____. Are the following functions odd, even or neither? How many zeros does the function appear to have?3143250190500488950019050016852902540003238504064000188. 189. 190.191.Analyzing Graphs and Tables of Polynomial Functions: Class Work398145031305400Identify any zeros (either as an integer or as an interval of x-values) of the function. Label any relative maximum and minimum.6985005778500192. 193.32131002667000488950000194. 195.xf(x)-25-110-11022314-1xf(x)-22-1-30-41-122354-2xf(x)-2-4-1002112-13-34-1196. 197. 198.Analyzing Graphs and Tables of Polynomial Functions: HomeworkIdentify any zeros (either as an integer or as an interval of x-values) of the function. Label any relative maximum and minimum.356235076835003517906794500199.200.419100850900037020508509000201. 202.xf(x)-22-14021-2203341xf(x)-26-120113213-140xf(x)-24-1-20-31-1213347203. 204. 205.Zeros and Roots of a Polynomial Function: Class Work509905028257500258445028257500For each graph below and its given degree, name the real zeros and their multiplicity, and state the number of imaginary zeros. 3492502730500206.207.208. 4th degree4th degree5th degreeName all of the real and imaginary zeros and state their multiplicity.209. fx=x+1x+2x+2x-3210. gx=x2-1(x2+1)211. y=x+12x+2(x-2)212. hx= x2x-10x+1213. y=x2-9x+32(x2+9)Zeros and Roots of a Polynomial Function: HomeworkFor each graph below and its given degree, name the real zeros and their multiplicity, and state the number of imaginary zeros. 349250139700004889500253990020955006921500214.215.216. 3rd degree4th degree6th degreeName all of the real and imaginary zeros and state their multiplicity.217. fx=x-1x+3x+3x-3218. gx=x2-4(x2+4)219. y=x+72(4x2-64)220. hx= x3x-7x-6x(2x+4)(x-5)221. y=x+42x2-16(x2+16)Zeros and Roots of a Polynomial Function by Factoring: Class WorkName all of the real and imaginary zeros and state their multiplicity.342265012890500222. fx=2x3+16x2+30x225. fx=x4-8x2-9223. fx=x4+9x2226. fx=2x3+x2-16x-15224. fx=2x3+3x2-8x-12227. fx=x3+4x2-25x-100228. Consider the function fx=x3+3x2-x-3. Use the fact that x+3 is a factor of f to factor this polynomial. Find the x-intercepts for the graph of f.At which x-values can the function change from being positive to negative or from negative to positive?For x<-3, is the graph above or below the x-axis? How can you tell?For -3<x<-1, is the graph above or below the x-axis? How can you tell? For -1<x<1, is the graph above or below the x-axis? How can you tell?For x>1, is the graph above or below the x-axis? How can you tell?Use the information generated in parts (f)–(i) to sketch a graph of f.36322008001000Zeros and Roots of a Polynomial Function by Factoring: HomeworkName all of the real and imaginary zeros and state their multiplicity.342265012890500229. fx=x3-3x2-2x+6232. fx=x4-x2-30230. fx=x4+x2-12233. fx=3x4-5x3+x2-5x-2231. fx=x3+5x2-9x-45234. fx=x4-5x3+20x-16235. Consider the function fx=x3-6x2-9x+14. Use the fact that x+2 is a factor of f to factor this polynomial. Find the x-intercepts for the graph of f.At which x-values can the function change from being positive to negative or from negative to positive?For x<-2, is the graph above or below the x-axis? How can you tell?For -2<x<1, is the graph above or below the x-axis? How can you tell? For 1<x<7, is the graph above or below the x-axis? How can you tell?For x>7, is the graph above or below the x-axis? How can you tell?307340019177000Use the information generated in parts (f)–(i) to sketch a graph of f.Writing Polynomials from Given Zeros: Class workWrite a polynomial function of least degree with integral coefficients that has the given zeros.2933700168910004051300168910236. -3, -2, 2240. 237. -3, -1, 2, 4 4051300125730238. ±3, 13, -5241. 239. 2, 3, i, -i, 35Writing Polynomials from Given Zeros: HomeworkWrite a polynomial function of least degree with integral coefficients that has the given zeros.2933700157480004260850201295242. 1, 2, 34246. 243. -1, 3, 0244. 0 mult. 2, -5, 1440055011239500245. -2i, 2i, -5(mult. 3)247.UNIT REVIEWMultiple ChoiceSimplify the following expression: 6p8q92p3q43-234pq3916p2q64pq3316p2q69The sides of a rectangle are (2x2 – 11x +1) ft and (3x – 4) ft find the perimeter of the rectangle.(2x2 – 8x – 3) ft(4x2 – 16x – 6) (5x3 – 11x – 3) ft(6x3 – 41x2 + 47x – 4) ft2The sides of a rectangle are (2x2 – 11x +1) ft and (3x – 4) ft find the area of the rectangle.(6x3 – 41x2 – 41x – 4) ft2 (6x3 – 25x2 + 47x – 4) ft2(6x3 – 41x2 + 47x – 4) ft2 (6x3 – 33x – 4) ft2A pool that is 10ft by 20 ft is going to have a deck (x) ft added all the way around the pool. Write an expression in simplified form for the area of the deck.60x+4x2ft230x+x2ft2200+60x+4x2ft2200+30x+x2ft2What is the area of a square with sides (6x – 2) inches?36x2-4 in2 36x2+4 in236x2-12x-4 in236x2-24x+4 in227w3x5-12w4x3+24w3x26w2x2 is equivalent to which of the following?9wx3-4w2x+4w39wx32-2w2x+4w9wx3-4w2x3+4w9wx3+4w2x+8w22a4-6a2+4÷a-22a3-3a-22a3-3a2-22a3+4a2-2a-4+-4a-22a3+4a2+2a+4+12a-2A box has volume of 3x2-2x-5 cm3 and a height of (x+1) cm. Find the area of the base of the box.(3x + 2) cm2 (3x – 2) cm2 (3x + 5) cm2 (3x – 5) cm2 2728913185738Using the graph, decide if the following function has an odd or even degree and the sign of the lead coefficient. odd degree; positiveodd degree; negativeeven degree; positiveeven degree; negativeWhich of the following equations is of an odd-function?y=3x5-2xy=5x7-3x3+9y=x5x7+x5y=7x10xf(x)-26-1A0213213-140What value should A be in the table so that the function has 4 zeros?-2013Name all of the real and imaginary zeros and state their multiplicity:y=x2+8x+16(4x2+64)Real zeros: -4 with multiplicity 2; Imaginary zeros: ± 4i each with multiplicity 1Real zeros: -4 with multiplicity 3, 4 with multiplicity 1; No imaginary zerosReal zeros: -4 with multiplicity 4; No imaginary zerosReal zeros: -4 with multiplicity 2; Imaginary zeros: 2i with multiplicity 237719007366000Extended ResponseGraph y=(x+2)2x+1xx-1x-3. Name the real zeros and their multiplicity.Given the function fx=3x3+3x2-6. Write the function in factored form. Name all of the real and imaginary zeros and state their multiplicity of the functionfx=x3-10x2+11x+70Write a polynomial function of least degree with integral coefficients that has the given zeros.-4.5, -1, 0, 1, 4.5Consider the graph of a degree 5 polynomial shown to the right, with x-intercepts -4, -2, 1, 3, and 5. Write an equation for a possible polynomial function that the graph represents.4470400217170000Answer Keyj6-64g9h616k69m2n4 64q69p2 80r15t2s8 34u2v18 27w3y38x3z33g16j1281h12 4k8n89m6 4p2q2 5s27t82r2 8u8v33 25w124z24 Yes, 5x2, degree: 2, monomial/quadratic, 5 Yes, -3y2+347y, degree: 2, binomial/quadratic, -3 Yes, 5a3-6a+3, degree: 3, trinomial/cubic, 5Not a polynomial functionNot a polynomial function6g2-4g-7-4t2+t+137x5+13x4+2x3+6x-1-12x3+6x2-6x-3 Perimeter = (6x2+12x-1) inches -8a3b+6a2b2+12a2b 35j4k3+63j3k2-14jk3+70jk2 8x2-8x-6 c3+4c2-3c-12 2m3-2m2-17m+15 12f3+22f2-18f+5 12t4-11t3+41t2-11t+9 Area = (30x2-23x-14) in.2 Area = π(63x3+186x2+160x+32) m2 Area = (2x2+12x+2) ft.2 Areadeck = (4x2+100x) ft.2 b2+4b+4 c2-2c+1 4d2+16de+16e2 25f2-81 (9x2+12x+4) in.232x5+320x4y+1280x3y2+2560x2y3+2560xy4+1024y5343a3+147a2b+21ab2+b3729x6-5832x5z+19440x4z2-34560x3z3+34560x2z4-18432xz5+4096z6y4-20y3z+150y2z2-500yz3+625z4Yes, 0.4x3+2x2, degree: 3, binomial/cubic, 0.4Not a polynomial functionYes, already in std form, degree: 4, no specific name/quartic, 11Yes, already in std form, degree: 2, trinomial/quadratic, 6/11Not a polynomial function-2n2-6n-3-3g3+10g2-5g-11-3x4+7x3+2x2-19x-166x3-7x2+24x-33Perimeter = (10x2+24x-10) inches 12x3-20x2-8x -18a3b+30a2b2+42ab 16j5k4+48j4k3-40j2k4+88j2k3 24x2+34x+5 8b2-40b+18 6c3+4c2-12c-8 6m3-27m2+22m+20 18f3+12f2-13f+4 2p4+16p3-p2-40p-10 15t4-19t3+29t2-15t+6 Area = (12x2-29x+15) in.2 Area = 81x3-27x+6) m2 Area = (3x2+33x+8) in.2 Areadeck = (4x2+146x+276) ft.2 9a2-1 b2-4b+4 c2-1 9d2-30de+25e2 25f2+90f+81 Area = (16x2-48xy+36y2) in.264a6-192a5b+240a4b2-160a3b3+60a2b4-12ab5+b627x3+54x2y+36xy2+8y33125y5-12500y4z+20000y3z2-16000y2z3+6400yz4-1024z5a4+28a3b+294a2b2+1372ab3+2401b43x2y(2xy – 1)5p2q(2p – 3pq – q)7m3(n3 – n2 + 2)(x – 8)(x + 3)(m – 3n)(m + 2n)(x – y)(x – y)(a + 4b)(a – 3b)(x – 4y)(x – 2y)(2x + 1)(x + 3)(3x – 2)(2x + 1)(5a – 3)(a + 4)(2m – n)(3m – n)(6p + 1)(p + 6)(2c + 5d)(2c + 5d)4x2y(2x-y)4mn3(2m2-m-8)3pq(-6p2q+1)(m - 6)(m + 4)(a - 12)(a - 1)(n + 3)(n - 2)(x – 7y)(x – 3y)(x + 9y)(x + 2y)(3x – 1)(2x – 1)(3p – 5)(5p + 1)(2m + 3)(5m – 1)(3x – y)(4x – y)(2p + 7)(2p + 5)(3m – 2n)(5m – n)5776335682(a – 1)(a2 + a + 1)(5x – 4y)(5x + 4y)(11a – 4b)(11a + 4b)(3x + 2y)(9x2 + 6xy + 4y2)(ab – c)(a2b2 + abc + c2)(2xy – 1)(2xy + 1)(x + 4)(2y + 5)(3m – 5)(3n – 1)(2x – 3)(y – 5)(5r + 3)(2s – 5)(2p – 1)(5q – 1)(5m + 3)(2n + 1)3x(x – 6)(x + 2)2m(3m – 1)(m + 1)3ab(a – 4)(a + 4)2x(3x + y)(9x2 – 3xy + y2)x2y(x + 10)(x + 2) (y + 3)(y2 – 3y + 9)(4m – 1)(16m2 + 4m + 1)(p – 6q)(p + 6q)(mn – 2)(mn + 2)Not Factorable(2x – 3y)(4x2 + 6xy + 9y2)(2m – 5)(3p – 1)(3x + 2)(2y + 5)(4r + 3)(s – 1)(3t – 1)(2r – 3)(4m + 3)(2n + 1)(x – 5)(3y – 4)3m(m – n)(m + n)-2x(3x – 1)(x + 5)2ab(3a – 5)(3a + 5)xy(x + 3)(x2 – 3x + 9)-3r(4r + 3)(r + 1)2xy(x – 1)(y – 1)3222773135162x2-x+32a2b2+4ab2-3b2x2 – 2x + 3 4a3+12a2+6a+9 + 13a-1a. 3x+14+24x-2 b. 24a. x2-4x+10-2x+1 b. -2Yes, because P(-1) = 0.B = (3t2-7t + 8 - 6t+1) cm.24x3-3x+6a3b+2a2b- 4a f+ 6f - 4f2 3x2-12x+45 - 133x+3 a. x2+2x+4-16x-2 b. -16 a. x2+4x-12+19x+1 b. 19 Yes, because P(1) = 0. 4f2-8f+16 - 32f+2height = 4t2+t+3+ 5t-1 cma. No b. YesOdd; positive; neitherEven; negative; evenEven; positive; neitherOdd; negative; neitherEven functionNeither NeitherOdd170: ∞, ∞ 171: ∞, -∞ 172: -∞, ∞ 173: ∞, -∞Odd function; 3 zerosEven function; 2 zerosNeither; 3 zerosEven function; 2 zerosOdd; negative; neitherEven; negative; evenEven; positive; evenOdd; negative; oddNeitherOdd functionOdd functionEven function184: ∞, ∞ 185: ∞, -∞ 186: ∞, -∞ 187: -∞,- ∞Even function; 2 zerosOdd function; 1 zeroNeither; 2 zerosOdd function; 1 zeroZeros: between x= -2 and x= -1, at x= 0, between x=1 and x= 2; relative max at x= -1; relative min at x=1Zeros: between x=-2 and x=-1, between x=-1 and x=0, between x=0 and x=1, between x=1 and x=2; relative max at x=-1 and x=1; relative min at x=0Zeros: at x=-2 and x=2; no relative max; relative min at x=0Zeros: between x=-2 and x=-1, between x=-1 and x=0 , at x=0, between x=0 and x=1, between x=1 and 2; relative max at x≈-.5 and x≈1.5; relative min at x≈-1.5 and x≈.5Zeros: between x=-1 and 0, at x=1, between x=3 and 4; relative max x=2; relative min at x=0 Zeros: at x=-1, between x=1 and 2; relative max at x=0; relative min at x=3Zeros: between x=-2 and x=-1, between x=1 and x=2, between x=3 and x=4; relative max at x=3; relative min at x=0 Zero: at x=2; no relative max or minZeros: at x≈-2, x≈-1, x≈0, x≈1,and x≈2; relative max at x=-1.5 and x=.5; relative min at x=-.5 and x=1.5Zeros: between x=-2 and x=-1, between x=1 and x=2; relative max at x=0; relative min at x=-1 and x=1No zeros; relative max at x=0; relative min at x=-1 and x=1Zeros: between x=2 and 3, and at x=4; relative max at x=1; relative min at x=0 and x=3Zeros: between x=0 and 1, at x=2; relative max at x=-1 and x=3; relative min at x=1Zeros: between x=-2 and x=-1, between x=1 and x=2; no relative max; relative min at x=0Real zeros: at x=-2 and x=2 ( both mult. of 2); no imaginary zerosReal zeros: at x=3 (mult. of 2); 2 imaginary zerosReal zeros: at x=-3, x = -1, x=3 (all mult. of 1), x=3 (mult. of 2); no imaginary zerosReal zeros: at x=-1 (mult. of 1), at x=-2 (mult. of 2) and x=3 (mult. of 1)Real zeros: at x=-1 (mult. of 1), at x=1 (mult. of 1); Imaginary zeros: at x= i (mult. of 1), at x=-i (mult. of 1)Real zeros: at x=-1 (mult. of 2), at x=2 (mult. of 1), at x=-2 (mult. of 1)Real zeros: at x=0 (mult. of 2), at x=10 (mult. of 1), at x=-1 (mult. of 1)Real zeros: at x=-3 (mult. of 3), at x=3 (mult. of 1); Imaginary zeros: at x=3i (mult. of 1), at x=-3i (mult. of 1)Real zeros: at x=-2 (mult. of 1) and at x=-1 (mult. of 1) and at x = 1 (mult. of 1); no imaginary zerosReal zeros: at x=-2 and x=2 (each mult. of 1); 2 imaginary zerosReal zeros: at x=-1.5 (mult. of 1) x=2 (mult. of 1) and at x=3 (mult. of 2); 2 imaginary zerosReal zeros: at x=1 (mult. of 1), at x=-3 (mult. of 2), at x=3 (mult. of 1)Real zeros: at x=2 (mult. of 1), at x=-2 (mult. of 1); Imaginary zeros: at x=2i (mult. of 1), at x=-2i (mult. of 1)Real zeros: at x=-7 (mult. of 2), x=4 (mult. of 1), at x=-4 (mult. of 1)Real zeros: at x=0 (mult. of 4), at x=7 (mult. of 1), at x=6 (mult. of 1), at x=-2 (mult. of 1), at x=5 (mult. of 1)Real zeros: at x=-4 (mult. of 3), at x=4 (mult. of 1); Imaginary zeros: at x=4i (mult. of 1), at x=-4i (mult. of 1) Real zeros: at x=0 (mult. of 1), at x=-3 (mult. of 1), at x=-5 (mult. of 1Real zeros: at x=0 (mult. of 2) 2 Imaginary zeros: at x= 3i (mult. of 1), at x=-3i (mult. of 1)Real zeros: at x=-1.5 (mult. of 1), at x= 2 (mult. of 1), at x=-2 (mult. of 1)Real zeros: at x=-3 (mult. of 1), at x=3 (mult. of 1);2 Imaginary zeros: at x= i (mult. of 1), at x=-i (mult. of 1)Real zeros: at x=-1 (mult. of 1), at x=-52 (mult. of 1), at x=3 (mult. of 1)Real zeros: at x=-5 (mult. of 1), at x=-4 (mult. of 1), at x=5 (mult. of 1)a. f(x) = (x + 3)(x + 1)(x – 1)-3, -1, 1-3, -1, 1Below, f(-4) is negative, OR since the degree is 3 and the leading coefficient is positive.Above, crosses at -3Below, crosses at -1Above, crosses at 13 Real zeros: at x=2 (mult. of 1), at x=-2 (mult. of 1), at x=3 (mult. of 1)Real zeros: at x=3 (mult. of 1), at x=-3 (mult. of 1);2 Imaginary zeros: at x=2i (mult. of 1), at x=-2i (mult. of 1)Real zeros: at x=-3 (mult. of 1), at x= 3 (mult. of 1), at x=-5 (mult. of 1)2 Real zeros: at x=6 (mult. of 1), at x=-6 (mult. of 1);2 Imaginary zeros: at x=i5 (mult. of 1), at x=-i5 (mult. of 1)Real zero: at x=2 (mult. of 1) and at x=-13 (mult. of 1); Imaginary zeros: at x=i (mult. of 1), at x=-i (mult. of 1)4 Real zeros: at x=1 (mult. of 1), at x=4 (mult. of 1), at x=-2 (mult. of 1), at x=2 (mult. of 1)a. f(x) = (x + 2)(x – 7)(x – 1)-2, 1, 7-2, 1 , 7Below, f(-3) is negative, or since the degree is 3 and the leading coefficient is positive. Above, crosses at -2Below, crosses at 1Above, crosses at 7fx=x+3x+2x-2fx=x+3x+1x-2x-4fx=x2-3x-13x+5fx=x-2(x-3)(x2+1)x-35fx=x(x-2)2fx=(x-1)2(x+1)2fx=x-1x-34x-2fx=xx+1x-3fx=x2x+5x-1 fx=(x2+4)x+53fx=x(x-1.5)(x+1.5)fx=x(x2-1)(x2-4)REVIEWDBCADBDDBAAA307340014922500x=-2 mult.of 2x=-1 mult.of 1x=0 mult.of 1x=1 mult.of 1x=3 mult.of 1 3(x-1)(x2+2x+2)x=-2 mult.of 1x=5 mult.of 1x=7 mult.of 14. fx=x(x2-1)(x2-20.25)5. f(x) = (x + 4)(x + 2)(x – 1)(x – 3)(x – 5) ................
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