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Cogito

Integrated Math 3 · Chapter 1 · Lesson 1

Practice Worksheet

Polynomial Behaviour

Degree and leading coefficient decide the ends.

12 problems · about 21 minutes · F-IF.C.7, A-APR.B.3

NameDate

Figure — use these to answer the problems

Make the leading coefficient negative and both ends swap. Count the turning points as you change b.
-8-8-6-6-4-4-2-222446688
y = 1x³ − 3x + 0
a, leading coefficient1
b-3
c0
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. A degree-4 polynomial. At most how many turning points?

    Answer

  2. What decides end behaviour?

    • The degree and the sign of the leading coefficient.
    • The constant term.

  3. f(x) = 3x⁴. How many ends go up?

    Answer

  4. f(x) = −3x⁴. How many ends go up?

    Answer

  5. f(x) = 2x⁵. How many ends go up?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. A degree-6 polynomial. At most how many turning points?

    Answer

  2. (x − 3)²(x + 1) = 0. Multiplicity of the root 3?

    Answer

  3. At an even multiplicity, does the graph cross the axis? 1 yes, 0 no.

    Answer

Stretch Yourself

Mixed problems. Work out what kind of question it is before you start.

4 problems

  1. Sort each polynomial by its end behaviour.

    Write each item under the heading it belongs to: Degree 2 · Degree 4 · Degree 3 · Degree 5

    Both ends the same way

    Ends go opposite ways

  2. A degree-3 polynomial. How many roots in total?

    Answer

  3. The Turns

    A degree-5 polynomial. At most how many turning points?

    Answer

  4. The Roots

    A degree-5 polynomial. How many roots, counting repeats and complex ones?

    Answer