5.3 The Fundamental Theorem of Algebra & Descartes’ Rule of Signs

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5.3 The Fundamental Theorem of Algebra & Descartes’ Rule of Signs Today’s Date: 12/18/17

Fundamental Theorem of Algebra Any polynomial of degree n has n roots. May be complex roots May have duplicates (multiplicity)

Ex 1) State the degree & list distinct zeros & their multiplicities. To find degree when in factored form, add exponents (total) Degree = 1 + 2 = 3 Zeros: 1 (mult. 1) –2 (mult. 2) Parenthesis → Opp sign Multiplicity → Exponent

T.O.O. Ex 2) State the degree & list distinct zeros & their multiplicities. 2 + 2 + 3 = 7 Zeros: 0 (mult. 2) –3 (mult. 2) ¾ (mult. 3)

Ex 3) Find poly. of least degree that is monic & zeros are 3 (mult. 2) and –1. Write in factored form. Monic: lead. coeff. = 1

Find poly. of least degree with zeros 1, –1, and 2 and P(0) = 4. Ex 4) Find poly. of least degree with zeros 1, –1, and 2 and P(0) = 4. Given to find leading coeff Plug in 0 for x and set = to 4 to find a

Descartes’ Rule of Signs Determines nature of the roots ~ the # of (+) or (–) real roots ~by counting # of sign changes

Ex 5) Use Descartes’ Rule of Signs to discuss the nature of the roots + Count sign changes change change # of (+) real roots: # of sign changes, or less by some multiple of 2 (SUBTRACT BY 2) (+) Real Roots: 2 sign changes → 2 or 0 (+) real roots (SUBTRACT BY 2)

Ex 5 cont.. (–) Real Roots: 1 sign change → 1 (–) real root To find (–) real roots: Plug in (–x) Count sign changes change (–) Real Roots: 1 sign change → 1 (–) real root Hint: Only odd exponents will change sign when plugging in –x Not or 0! Why?

Example 5 cont… Summarize your results in table form (+) (–) i 2 1 1 2 2 or 0 (+) real roots; 1 (–) real root Summarize your results in table form (+) (–) i Deg = 3 → 3 zeros (each row must add to 3) 2 1 1 2 2 scenarios

T.O.O. Ex 6) Use Descartes’ Rule of Signs to discuss the nature of the roots 2 or 0 (+) real roots + 2 or 0 (–) real roots (+) (–) i 2 2 Deg = 4 → 4 zeros 2 2 2 2 4 4 scenarios

Homework #505 Pg. 293 1 – 25 odd

Intermediate Value Theorem If values of P(x) change from (+) to (–) or (–) to (+), there must be a zero in between. P(x) is (+) Zero in between (it crosses the x-axis!) P(x) is (–)

Ex 4) Use I.V.T. & synthetic division to show that P(x) has a zero between -3 & -4 1 1 -9 6 ↓ -3 6 9 (+) 1 -2 -3 15 Changes signs, so must be a 0 between -4 1 1 -9 6 ↓ -4 12 -12 1 -3 3 -6 (–)

FYI – the Real Graph of Zoomed in to see zero