Bell ringer Simplify the following expressions as much as you can! 𝑥 4 − 3𝑥 2 + 2𝑥 4 − 5𝑥 3 +4 3𝑥 3 + 7𝑥 2 + 4𝑥 4 + 𝑥 3 −20 Write your final answers in standard form (think of standard form for quadratics!)

Polynomials October 27, 2015

Parts of a Polynomial DEGREES 2𝑥 4 − 5𝑥 3 +4 TERMS

Degrees of a polynomial The power of x helps you classify the polynomial and graphs. Look at the highest power! 𝑥 2 - squared 𝑥 3 - CUBIC Think of your prefixes for the next ones… 𝑥 4 - 𝑥 5 - QUARTIC!! QUINTIC!!

Standard form Combines any like terms Arranges the terms of polynomial by their degree from greatest to least 𝑥 4 − 3𝑥 2 + 2𝑥 4 − 5𝑥 3 +4 3𝑥 4 − 3𝑥 2 − 5𝑥 3 +4 Combine like terms 3𝑥 4 − 5𝑥 3 − 3𝑥 2 +4 write degrees in order

Write each polynomial in standard form and classify it 3𝑥+ 9𝑥 2 +5 4𝑥− 6𝑥 2 + 𝑥 4 + 10𝑥 2 −12 3− 4𝑥 5 + 2𝑥 2 + 3𝑥 5 +10 9𝑥 2 +3𝑥+5;𝑑𝑒𝑔𝑟𝑒𝑒 𝑜𝑓 2;𝑞𝑢𝑎𝑑𝑟𝑎𝑡𝑖𝑐 𝑥 4 + 4𝑥 2 +4𝑥−12 𝑑𝑒𝑔𝑟𝑒𝑒 𝑜𝑓 4;𝑞𝑢𝑎𝑟𝑡𝑖𝑐 −𝑥 5 + 2𝑥 2 +13 𝑑𝑒𝑔𝑟𝑒𝑒 𝑜𝑓 5;𝑞𝑢𝑖𝑛𝑡𝑖𝑐

End behaviors 𝑦= −𝑥 2 +6𝑥 𝑦= −0.3𝑥 3 +4𝑥+2 𝑦= 𝑥 4 − 3𝑥 3 +5𝑥 𝑦= 𝑥 3 Look at the graphs of the following equations. Predict and conclude the rule for end behavior 𝑦= −0.3𝑥 3 +4𝑥+2 𝑦= 𝑥 4 − 3𝑥 3 +5𝑥 𝑦= 𝑥 3 𝑦= −2𝑥 4 + 8𝑥 3 − 8𝑥 2 +2

End behavior rule If the degree is even: ________________________________ If the degree is odd: _________________________________ If the leading coefficient is negative: ___________________ Each end goes in the same direction Each end goes in the opposite direction If degree is even, both ends go down If odd, left end up, right end down

Maximum number of turns 𝑦= −𝑥 2 +6𝑥 Look at the graphs of the following equations. Predict and conclude the rule for the maximum number of turns. 𝑦= −0.3𝑥 3 +4𝑥+2 𝑦= 𝑥 4 − 3𝑥 3 +5𝑥 𝑦= 𝑥 3 𝑦= −2𝑥 4 + 8𝑥 3 − 8𝑥 2 +2

Maximum number of turns rule What did you come up with? Max # of turns: Highest degree – 1

Partner Work/Homework Workbook page 47 #1, 2, 4, 8-11