<p>Pre-Calc Worksheet: Real Zeros of Polynomials Do these problems by hand.</p><p>Name______Period______</p><p>1. List all possible rational zeros given by the Rational Root Test (but don’t check to see which actually are zeros). a. f( x )= 2 x5 + 3 x 3 + 4 x 2 - 8 b. p( x )= x4 - 3 x 3 - 6 x + 20</p><p>2. Use Descartes’ Rule of Signs to determine how many positive and how many negative real zeros the polynomial can have. a. f( x )= x3 - x 2 - x - 3 positive: ______</p><p> negative: ______</p><p> b. f( x )= 2 x6 + 5 x 4 - x 3 - 5 x - 1 positive: ______</p><p> negative: ______</p><p> c. f( x )= x5 + 4 x 3 - x 2 + 6 x positive: ______</p><p> negative: ______</p><p>3. Find integers that are upper and lower bounds for the real zeros of the polynomial. Also use the Intermediate Value Theorem to identify consecutive integers between which any real zeros reside. Also use Descartes’ Rule of Signs. a. p( x )= x3 - 3 x 2 + 4 UB: ______</p><p>LB: ______</p><p># poss. (+) zeros: ______</p><p># poss. (-) zeros: ______</p><p> zeros will be between: ______b. . p( x )= x4 - 2 x 3 + x 2 - 9 x + 2 UB: ______</p><p>LB: ______</p><p># poss. (+) zeros: ______</p><p># poss. (-) zeros: ______</p><p> zeros will be between: ______</p><p> c. . p( x )= x5 - x 4 + 1 UB: ______</p><p>LB: ______</p><p># poss. (+) zeros: ______</p><p># poss. (-) zeros: ______</p><p> zeros will be between: ______</p><p>For 4-12, a) state the number of possible positive and negative real roots. B. List the possible rational roots. C. Then find all real zeros of the polynomial.</p><p>4. f( x )= x3 - 3 x - 2 # poss. (+) zeros: ______</p><p># poss. (-) zeros: ______</p><p> p : ______q</p><p>Real zeros: ______5. f( x )= x3 - 6 x 2 + 12 x - 8 # poss. (+) zeros: ______</p><p># poss. (-) zeros: ______</p><p> p : ______q</p><p>Real zeros: ______</p><p>6. f( x )= x3 - 4 x 2 + x + 6 # poss. (+) zeros: ______</p><p># poss. (-) zeros: ______</p><p> p : ______q</p><p>Real zeros: ______7. f( x )= x3 + 4 x 2 + 3 x - 2 # poss. (+) zeros: ______</p><p># poss. (-) zeros: ______</p><p> p : ______q</p><p>Real zeros: ______</p><p>8. f( x )= x4 - 6 x 3 + 4 x 2 + 15 x + 4 # poss. (+) zeros: ______</p><p># poss. (-) zeros: ______</p><p> p : ______q</p><p>Real zeros: ______9. f( x )= x3 - 3 x 2 - 4 x + 12 # poss. (+) zeros: ______</p><p># poss. (-) zeros: ______</p><p> p : ______q</p><p>Real zeros: ______</p><p>10. f( x )= 2 x4 + 3 x 3 - 4 x 2 - 3 x + 2 # poss. (+) zeros: ______</p><p># poss. (-) zeros: ______</p><p> p : ______q</p><p>Real zeros: ______11. f( x )= x4 - 5 x 3 + 6 x 2 + 4 x - 8 # poss. (+) zeros: ______</p><p># poss. (-) zeros: ______</p><p> p : ______q</p><p>Real zeros: ______</p><p>12. f( x )= x5 - x 4 - 3 x 3 + 5 x 2 - 2 x # poss. (+) zeros: ______</p><p># poss. (-) zeros: ______</p><p> p : ______q</p><p>Real zeros: ______</p>