Name: ______Mods: ______Date: ______

5.2 Discovery Activity

  1. Think of two numbers, a and b, that have a product of zero and write them below.
    a = ______b = ______ab = ______
  2. Think of another combination of numbers, a and b, that also have a product of zero and write them below.
    a = ______b = ______ab = ______
  3. Can you think of two numbers, not including zero, that have a product of zero? Why or why not. Explain using a complete sentence.
    ______
  4. Use what you discovered above to finish the following statement:
    If ab = 0, then either a= _____ or b = _____.
    This is called the ZERO PRODUCT PROPERTY. We learned this last unit, but now have a refresher on this concept.
  5. Consider the polynomial for the following questions.
  6. Classify this polynomial according to its degree and number of terms.
  7. Factor the polynomial. Hint: Always look for this first.
    Note: These are called theLINEAR FACTORS of the polynomial.
  8. Using the factored form of the polynomial from #5band the zero product property discovered in #4, find the values of x that satisfy . Hint: You should have three different values of x.
    The values of x such that are called the ZEROS, ROOTS, SOLUTIONS, orX-INTERCEPTS of the polynomial. This is just like we talked about last unit and it applies to polynomials with degree larger than 2 as well.
  9. What do you expect the end behavior of to be?
  10. Draw a rough sketch on the coordinate plane below of what the polynomial will look like now that we’ve found the x-intercepts (part b) and considered its end behavior (part d). Note: This will be a very rough sketch since we are only using a limited amount of information.
  11. Now, take out your chromebook (only one person in your group needs to do this) and go to desmos.com. Click launch calculator and type in the polynomial . To enter the exponents, use the ^ key on your keyboard. What does the graph of this polynomial actually look like? Draw it below by plotting several specific points.
  12. Was your rough sketch very good/accurate? Explain in complete sentences.
    ______
  13. Brainstorm some possible ways that you could have made your rough sketch more accurate without using technology. List them below.
    ______
  14. Working Backwards
    You are given only the zeros of a polynomial function. We do not know its equation and consequently do not know what it looks like when graphed.
  15. Suppose the zeros of a polynomial function f(x) are -2, 2, and 3.Applying what we know about how the factors of a polynomial are related to its zeros and its equation (5b & 5c), what are the three linear factors of the polynomial.
  16. Expand this polynomial using the distributive property (FOIL) to determine the cubic polynomial equation that fits the given criteria. Write the polynomial equation in standard form.
  17. Now suppose the zeros of a different quartic polynomial function g(x) are -2, -2, 2, and 3. What is the equation for this polynomial in standard form?
  18. Graph both functions (using desmos) and sketch them below. Label the graph with the polynomial that corresponds to it.
  19. How do the graphs differ? How are they similar? Explain in complete sentences.
    ______

The reason the graph of these polynomial functions differ despite having the same zeros is because the linear factor x + 2 appears twice in the polynomial g(x). This means that -2 is a MULTIPLE ZERO.

In fact, since the linear factor x + 2 appears twice in g(x), we say that
-2 is a zero of MULTIPLICITY2 of g(x). In general, multiplicity means that a factor appears more than once.

How Multiple Zeros Affect a Graph
- If a is a zero of even multiplicity, the graph touches the x-axis at a
and ______.
- If a is a zero of odd multiplicity, the graph ______the x-
axis at a.

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