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Introduction To Polynomials

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Presentation on theme: "Introduction To Polynomials"— Presentation transcript:

1 Introduction To Polynomials

2 The following are examples of monomials in one variable

3 The following are examples of monomials in one variable

4 The following are examples of monomials in one variable

5 The following are examples of monomials in one variable

6 The following are examples of monomials in one variable

7 The following are examples of monomials in one variable

8 A monomial is an expression of the type axn,

9 A monomial is an expression of the type axn,
where a is a real number constant

10 A monomial is an expression of the type axn,
where a is a real number constant and n is a nonnegative integer.

11 Algebraic expressions like the following are polynomials.

12 Algebraic expressions like the following are polynomials.

13 Algebraic expressions like the following are polynomials.

14 Algebraic expressions like the following are polynomials.

15 Algebraic expressions like the following are polynomials.

16 Algebraic expressions like the following are polynomials.

17 Algebraic expressions like the following are polynomials.

18 Algebraic expressions like the following are polynomials.

19 Algebraic expressions like the following are polynomials.

20 Algebraic expressions like the following are polynomials.

21 A polynomial is a monomial or a combination of sums and/or differences of monomials.

22 The following algebraic expressions are not polynomials.

23 The following algebraic expressions are not polynomials.

24 The following algebraic expressions are not polynomials.

25 The following algebraic expressions are not polynomials.

26 The following algebraic expressions are not polynomials.
Polynomials do not contain quotients.

27 Evaluate the polynomial when x = 2

28 Evaluate the polynomial when x = 2

29 Evaluate the polynomial when x = 2

30 Evaluate the polynomial when x = 2

31 Evaluate the polynomial when x = 2

32 Evaluate the polynomial when x = 2

33 Evaluate the polynomial when x = 2

34 Evaluate the polynomial when x = 2

35 Evaluate the polynomial when x = 2

36 Evaluate the polynomial when x = 2

37 Evaluate the polynomial when x = 2

38 Evaluate the polynomial when x = 2

39 Evaluate the polynomial when x = 2

40 Evaluate the polynomial when x = 2

41 Evaluate the polynomial when x = 2

42 Evaluate the polynomial when x = 2

43 Evaluate the polynomial when x = 2

44 Evaluate the polynomial when x = 2

45 Evaluate the polynomial when x = 2

46 Evaluate the polynomial when x = 2

47 Evaluate the polynomial when x = 2

48 Evaluate the polynomial when x = 2

49 Evaluate the polynomial when x = 2

50 Evaluate the polynomial when x = 2

51 Evaluate the polynomial when x = -4

52 Evaluate the polynomial when x = -4

53 Evaluate the polynomial when x = -4

54 Evaluate the polynomial when x = -4

55 Evaluate the polynomial when x = -4

56 Evaluate the polynomial when x = -4

57 Evaluate the polynomial when x = -4

58 Evaluate the polynomial when x = -4

59 Evaluate the polynomial when x = -4

60 Evaluate the polynomial when x = -4

61 Evaluate the polynomial when x = -4

62 Evaluate the polynomial when x = -4

63 Evaluate the polynomial when x = -4

64 Evaluate the polynomial when x = -4

65 Evaluate the polynomial when x = -4

66 Evaluate the polynomial when x = -4

67 Evaluate the polynomial when x = -4

68 Evaluate the polynomial when x = -4

69 Evaluate the polynomial when x = -4

70 Evaluate the polynomial when x = -4

71 Evaluate the polynomial when x = -4

72 x Use only the given graph of y = 2x – 2 to evaluate the polynomial

73 x Use only the given graph of y = 2x – 2 to evaluate the polynomial
when x = 3 x

74 x Use only the given graph of y = 2x – 2 to evaluate the polynomial
when x = 3 x

75 x Use only the given graph of y = 2x – 2 to evaluate the polynomial
when x = 3 x

76 x Use only the given graph of y = 2x – 2 to evaluate the polynomial
when x = 3 x

77 x Use only the given graph of y = 2x – 2 to evaluate the polynomial
when x = 3 x

78 x Use only the given graph of y = 2x – 2 to evaluate the polynomial
when x = 3 x

79

80 In a sports league of x teams in which each team plays every other team twice, the total number of games N to be played is given by the polynomial equation

81 In a sports league of x teams in which each team plays every other team twice, the total number of games N to be played is given by the polynomial equation N = x2 – x

82 In a sports league of x teams in which each team plays every other team twice, the total number of games N to be played is given by the polynomial equation N = x2 – x A women’s slow-pitch softball league has 10 teams.

83 In a sports league of x teams in which each team plays every other team twice, the total number of games N to be played is given by the polynomial equation N = x2 – x A women’s slow-pitch softball league has 10 teams. What is the total number of games to be played?

84 Evaluate the polynomial when x = 10

85 Evaluate the polynomial when x = 10

86 Evaluate the polynomial when x = 10

87 Evaluate the polynomial when x = 10

88 Evaluate the polynomial when x = 10

89 Evaluate the polynomial when x = 10

90 Evaluate the polynomial when x = 10

91 Evaluate the polynomial when x = 10

92 Evaluate the polynomial when x = 10

93 Evaluate the polynomial when x = 10

94 The concentration C, in parts per million of a certain antibiotic in the bloodstream after t hours is given by the polynomial expression

95 The concentration C, in parts per million of a certain antibiotic in the bloodstream after t hours is given by the polynomial expression C = -0.05t2 + 2t + 2

96 The concentration C, in parts per million of a certain antibiotic in the bloodstream after t hours is given by the polynomial expression C = -0.05t2 + 2t + 2 To find the concentration after 2 hours, we evaluate the polynomial when t = 2

97 Evaluate the polynomial when t = 0

98 Evaluate the polynomial when t = 0

99 Evaluate the polynomial when t = 0

100 Evaluate the polynomial when t = 0

101 Evaluate the polynomial when t = 0

102 Evaluate the polynomial when t = 0

103 Evaluate the polynomial when t = 0

104 Evaluate the polynomial when t = 0

105 Evaluate the polynomial when t = 0

106 Evaluate the polynomial when t = 0

107 Evaluate the polynomial when t = 0

108 Evaluate the polynomial when t = 0

109 Evaluate the polynomial when t = 0

110 Evaluate the polynomial when t = 0

111 Evaluate the polynomial when t = 0

112 Evaluate the polynomial when t = 0

113 Evaluate the polynomial when t = 0

114 Evaluate the polynomial when t = 2

115 Evaluate the polynomial when t = 2

116 Evaluate the polynomial when t = 2

117 Evaluate the polynomial when t = 2

118 Evaluate the polynomial when t = 2

119 Evaluate the polynomial when t = 2

120 Evaluate the polynomial when t = 2

121 Evaluate the polynomial when t = 2

122 Evaluate the polynomial when t = 2

123 Evaluate the polynomial when t = 2

124 Evaluate the polynomial when t = 2

125 Evaluate the polynomial when t = 10

126 Evaluate the polynomial when t = 10

127 Evaluate the polynomial when t = 10

128 Evaluate the polynomial when t = 10

129 Evaluate the polynomial when t = 10

130 Evaluate the polynomial when t = 10

131 Evaluate the polynomial when t = 10

132 Evaluate the polynomial when t = 10

133 Evaluate the polynomial when t = 10

134 Evaluate the polynomial when t = 10

135 Evaluate the polynomial when t = 10

136 Evaluate the polynomial when t = 20

137 Evaluate the polynomial when t = 20

138 Evaluate the polynomial when t = 20

139 Evaluate the polynomial when t = 20

140 Evaluate the polynomial when t = 20

141 Evaluate the polynomial when t = 20

142 Evaluate the polynomial when t = 20

143 Evaluate the polynomial when t = 20

144 Evaluate the polynomial when t = 20

145 Evaluate the polynomial when t = 20

146 Evaluate the polynomial when t = 20

147 Evaluate the polynomial when t = 30

148 Evaluate the polynomial when t = 30

149 Evaluate the polynomial when t = 30

150 Evaluate the polynomial when t = 30

151 Evaluate the polynomial when t = 30

152 Evaluate the polynomial when t = 30

153 Evaluate the polynomial when t = 30

154 Evaluate the polynomial when t = 30

155 Evaluate the polynomial when t = 30

156 Evaluate the polynomial when t = 30

157 Evaluate the polynomial when t = 30

158 The polynomial in the previous example can be graphed if we evaluate the polynomial for several values of t. We list the values in a table and show the graph below.

159 The concentration peaks when x = 20.
After 40 hours the concentration is 0.

160 Find an equivalent polynomial using only addition.

161 Find an equivalent polynomial using only addition.

162 Find an equivalent polynomial using only addition.

163 Find an equivalent polynomial using only addition.

164 Find an equivalent polynomial using only addition.

165 Find an equivalent polynomial using only addition.

166 Find an equivalent polynomial using only addition.

167 Find an equivalent polynomial using only addition.

168 Find an equivalent polynomial using only addition.

169 Find an equivalent polynomial using only addition.

170 Find an equivalent polynomial using only addition.

171 Identify the terms of the polynomial

172 Identify the terms of the polynomial

173 Identify the terms of the polynomial

174 Identify the terms of the polynomial

175 Identify the terms of the polynomial

176 Identify the terms of the polynomial

177 Identify the terms of the polynomial

178 Identify the terms of the polynomial

179 Identify the terms of the polynomial

180 Identify the terms of the polynomial

181 Identify the terms of the polynomial

182 Like Terms Not Like Terms

183 Like Terms Not Like Terms

184 Like Terms Not Like Terms

185 Like Terms Not Like Terms

186 Like Terms Not Like Terms

187 Like Terms Not Like Terms

188 Like Terms Not Like Terms

189 Like Terms Not Like Terms

190 Like Terms Not Like Terms

191 Like Terms Not Like Terms

192 Like Terms Not Like Terms

193 Like Terms Not Like Terms

194 Like Terms Not Like Terms

195 When you add like terms you get like terms

196 Like Terms Not Like Terms

197 Like Terms Not Like Terms

198 Like Terms Not Like Terms

199 Like Terms Not Like Terms

200 Like Terms Not Like Terms

201 Like Terms Not Like Terms

202 Identify the like terms of the polynomial

203 Identify the like terms of the polynomial

204 Identify the like terms of the polynomial

205 Identify the like terms of the polynomial

206 Identify the like terms of the polynomial

207 Identify the like terms of the polynomial

208 Identify the like terms of the polynomial

209 Identify the like terms of the polynomial

210 Identify the like terms of the polynomial

211 Identify the like terms of the polynomial

212 Identify the like terms of the polynomial

213 Identify the like terms of the polynomial

214 Identify the like terms of the polynomial

215 Identify the like terms of the polynomial

216 Identify the like terms of the polynomial

217 Identify the like terms of the polynomial

218 Identify the like terms of the polynomial

219 Identify the like terms of the polynomial

220 Identify the like terms of the polynomial

221 Identify the like terms of the polynomial

222 Coefficients What is the coefficient of the term?

223 Coefficients What is the coefficient of the term?

224 Coefficients Identify the coefficients

225 Coefficients Identify the coefficients

226 Coefficients Identify the coefficients

227 Coefficients Identify the coefficients

228 Coefficients Identify the coefficients

229 Coefficients Identify the coefficients

230 Coefficients Identify the coefficients

231 Coefficients Identify the coefficients

232 Coefficients Identify the coefficients

233 Coefficients Identify the coefficients

234 Coefficients Identify the coefficients

235 Collect the like terms

236 Collect the like terms

237 Collect the like terms

238 Collect the like terms

239 Collect the like terms

240 Collect the like terms

241 Collect the like terms

242 Collect the like terms

243 Collect the like terms

244 Collect the like terms

245 Collect the like terms

246 Collect the like terms

247 Collect the like terms

248 Collect the like terms

249 Collect the like terms

250 Collect the like terms

251 Collect the like terms

252 Collect the like terms

253 Collect the like terms

254 Collect the like terms

255 Collect the like terms

256 Collect the like terms

257 Collect the like terms

258 Collect the like terms

259 Collect the like terms

260 Collect the like terms

261 Collect the like terms

262 Collect the like terms

263 Collect the like terms

264 Collect the like terms

265 Collect the like terms

266 Collect the like terms

267 Collect the like terms

268 Collect the like terms

269 Collect the like terms

270 Collect the like terms.

271 Collect the like terms.

272 Collect the like terms.

273 Collect the like terms.

274 Collect the like terms.

275 Collect the like terms.

276 Collect the like terms.

277 Collect the like terms.

278 Collect the like terms.

279 Collect the like terms.

280 Collect the like terms.

281 Collect the like terms.

282 Collect the like terms.

283 Collect the like terms.

284 Collect the like terms.

285 Collect the like terms.

286 Collect the like terms.

287 Collect the like terms.

288 Collect the like terms.

289 Collect the like terms.

290 Collect the like terms.

291 Collect the like terms.

292 Descending order

293 Descending order

294 Descending order

295 Descending order

296 Descending order

297 Descending order

298 Arrange the polynomial in descending order

299 Arrange the polynomial in descending order

300 Arrange the polynomial in descending order

301 Arrange the polynomial in descending order

302 Arrange the polynomial in descending order

303 Arrange the polynomial in descending order

304 Arrange the polynomial in descending order

305 Arrange the polynomial in descending order

306 Arrange the polynomial in descending order

307 Arrange the polynomial in descending order

308 Arrange the polynomial in descending order

309 Collect like terms and arrange the polynomial in descending order

310 Collect like terms and arrange the polynomial in descending order

311 Collect like terms and arrange the polynomial in descending order

312 Collect like terms and arrange the polynomial in descending order

313 Collect like terms and arrange the polynomial in descending order

314 Collect like terms and arrange the polynomial in descending order

315 Collect like terms and arrange the polynomial in descending order

316 Collect like terms and arrange the polynomial in descending order

317 Collect like terms and arrange the polynomial in descending order

318 Collect like terms and arrange the polynomial in descending order

319 The degree of a term is the exponent of the variable.

320 The degree of a term is the exponent of the variable.
What is the degree of the term?

321 The degree of a term is the exponent of the variable.
What is the degree of the term?

322 The degree of a term is the exponent of the variable.
What is the degree of the term?

323 Identify the degree of each term of

324 Identify the degree of each term of

325 Identify the degree of each term of

326 Identify the degree of each term of

327 Identify the degree of each term of

328 Identify the degree of each term of

329 Identify the degree of each term of

330 Identify the degree of each term of

331 The degree of a polynomial is the largest of the degrees of the terms, unless it is the polynomial 0.

332 The polynomial 0 is a special case.

333 The polynomial 0 is a special case.

334 The polynomial 0 is a special case.
We agree that it has no degree either as a term or as a polynomial.

335 The polynomial 0 is a special case.
We agree that it has no degree either as a term or as a polynomial. This is because we can express

336 The polynomial 0 is a special case.
We agree that it has no degree either as a term or as a polynomial. This is because we can express 0 as 0x5 = 0x7,

337 The polynomial 0 is a special case.
We agree that it has no degree either as a term or as a polynomial. This is because we can express 0 as 0x5 = 0x7, and so on, using any exponent we wish.

338 Identify the degree of the polynomial

339 Identify the degree of the polynomial

340 Identify the degree of the polynomial

341 Identify the degree of the polynomial

342 Identify the degree of the polynomial

343 Identify the degree of the polynomial

344 Identify the degree of the polynomial

345 Identify the degree of the polynomial

346 Identify the degree of the polynomial

347 Lets summarize the terminology that we have learned
Lets summarize the terminology that we have learned. Using the polynomial

348 Term Coefficient Degree of the Term Degree of the Polynomial

349 Term Coefficient Degree of the Term Degree of the Polynomial

350 Term Coefficient Degree of the Term Degree of the Polynomial

351 Term Coefficient Degree of the Term Degree of the Polynomial

352 Term Coefficient Degree of the Term Degree of the Polynomial

353 Term Coefficient Degree of the Term Degree of the Polynomial

354 Term Coefficient Degree of the Term Degree of the Polynomial

355 Term Coefficient Degree of the Term Degree of the Polynomial

356 Term Coefficient Degree of the Term Degree of the Polynomial

357 Term Coefficient Degree of the Term Degree of the Polynomial

358 Term Coefficient Degree of the Term Degree of the Polynomial

359 Term Coefficient Degree of the Term Degree of the Polynomial

360 Term Coefficient Degree of the Term Degree of the Polynomial

361 Term Coefficient Degree of the Term Degree of the Polynomial

362 Term Coefficient Degree of the Term Degree of the Polynomial

363 Term Coefficient Degree of the Term Degree of the Polynomial

364 Missing terms. Identify the missing terms in the polynomial

365 Missing terms. Identify the missing terms in the polynomial

366 Missing terms. Identify the missing terms in the polynomial

367 Missing terms. Identify the missing terms in the polynomial

368 Missing terms. Identify the missing terms in the polynomial

369 Missing terms. Identify the missing terms in the polynomial

370 Missing terms. Identify the missing terms in the polynomial

371 Missing terms. Write the polynomial with its missing terms

372 Missing terms. Write the polynomial with its missing terms

373 Missing terms. Write the polynomial with its missing terms

374 Missing terms. Write the polynomial with its missing terms

375 Missing terms. Write the polynomial with its missing terms

376 Missing terms. Write the polynomial with its missing terms

377 Missing terms. Write the polynomial with its missing terms

378 Missing terms. Write the polynomial with its missing terms

379 Missing terms. Write the polynomial with its missing terms

380 Missing terms. Write the polynomial with its missing terms

381 Missing terms. Write the polynomial with its missing terms

382 Missing terms. Write the polynomial with its missing terms

383 Missing terms. Write the polynomial with its missing terms

384 Classifying Polynomials

385 Classifying Polynomials
Polynomials with one term are called monomials.

386 Classifying Polynomials
Polynomials with one term are called monomials. Polynomials with two terms are called binomials.

387 Classifying Polynomials
Polynomials with one term are called monomials. Polynomials with two terms are called binomials. Polynomials with three terms are called trinomials.

388 Classifying Polynomials
Polynomials with one term are called monomials. Polynomials with two terms are called binomials. Polynomials with three terms are called trinomials. MONOMIALS BINOMIALS TRINOMIALS NONE OF THESE

389 Classifying Polynomials
Polynomials with one term are called monomials. Polynomials with two terms are called binomials. Polynomials with three terms are called trinomials. MONOMIALS BINOMIALS TRINOMIALS NONE OF THESE

390 Classifying Polynomials
Polynomials with one term are called monomials. Polynomials with two terms are called binomials. Polynomials with three terms are called trinomials. MONOMIALS BINOMIALS TRINOMIALS NONE OF THESE

391 Classifying Polynomials
Polynomials with one term are called monomials. Polynomials with two terms are called binomials. Polynomials with three terms are called trinomials. MONOMIALS BINOMIALS TRINOMIALS NONE OF THESE

392 Classifying Polynomials
Polynomials with one term are called monomials. Polynomials with two terms are called binomials. Polynomials with three terms are called trinomials. MONOMIALS BINOMIALS TRINOMIALS NONE OF THESE

393 Classifying Polynomials
Polynomials with one term are called monomials. Polynomials with two terms are called binomials. Polynomials with three terms are called trinomials. MONOMIALS BINOMIALS TRINOMIALS NONE OF THESE

394 Classifying Polynomials
Polynomials with one term are called monomials. Polynomials with two terms are called binomials. Polynomials with three terms are called trinomials. MONOMIALS BINOMIALS TRINOMIALS NONE OF THESE

395 Classifying Polynomials
Polynomials with one term are called monomials. Polynomials with two terms are called binomials. Polynomials with three terms are called trinomials. MONOMIALS BINOMIALS TRINOMIALS NONE OF THESE

396 Classifying Polynomials
Polynomials with one term are called monomials. Polynomials with two terms are called binomials. Polynomials with three terms are called trinomials. MONOMIALS BINOMIALS TRINOMIALS NONE OF THESE

397 Classifying Polynomials
Polynomials with one term are called monomials. Polynomials with two terms are called binomials. Polynomials with three terms are called trinomials. MONOMIALS BINOMIALS TRINOMIALS NONE OF THESE

398 Classifying Polynomials
Polynomials with one term are called monomials. Polynomials with two terms are called binomials. Polynomials with three terms are called trinomials. MONOMIALS BINOMIALS TRINOMIALS NONE OF THESE

399 Classifying Polynomials
Polynomials with one term are called monomials. Polynomials with two terms are called binomials. Polynomials with three terms are called trinomials. MONOMIALS BINOMIALS TRINOMIALS NONE OF THESE

400 Classifying Polynomials
Polynomials with one term are called monomials. Polynomials with two terms are called binomials. Polynomials with three terms are called trinomials. MONOMIALS BINOMIALS TRINOMIALS NONE OF THESE

401 Lesson 4.3 Numbers

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