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Activity 30:

Activity 30:. Polynomial Functions (Section 4.1, pp. 310-322) Date: and Their Graphs. Polynomial Functions:. A polynomial function of degree n is a function of the form.

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Activity 30:

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  1. Activity 30: Polynomial Functions (Section 4.1, pp. 310-322) Date: and Their Graphs

  2. Polynomial Functions: A polynomial function of degree n is a function of the form where n is a non-negative integer and an ≠ 0. The numbers a0, a1, . . ., an are called the coefficients of the polynomial. The number a0 is the constant coefficient or constant term. The number an, the coefficient of the highest power, is the leading coefficient, and the term anxn is the leading term.

  3. End Behavior and the Leading Term:

  4. Real Zeros of Polynomials: If P(x) is a polynomial and c is a real number, then the following are equivalent: c is a zero of P(x); P(c) = 0; x − c is a factor of P(x); (c,0) is an x-intercept of the graph of y = P(x).

  5. Example 1: Determine the end behavior of the polynomial P(x) = 3(x2 − 4)(x − 1)3. ? Consequently, multiply the above out one has a degree 5 polynomial with leading term 3x5 So y → +∞ as x → +∞ and y → −∞ as x → −∞

  6. Intermediate Value Theorem for Polynomials: If P(x) is a polynomial function and P(a) and P(b) have opposite signs, then there exists at least one value c between a and b for which P(c) = 0.

  7. Example 2: Use the Intermediate Value Theorem to show that the polynomial P(x) = −x4+3x3−2x+1 has a zero in the interval [2, 3]. Since P(2)>0 and P(3)< 0 there must be some number c between 2 and 3 such that P(c) = 0.

  8. Example 3: Let P(x) = (x + 2)(x − 1)(x − 3). Find the zeros of P(x) and sketch its graph.

  9. Example 4: Let P(x) = −2x4 − x3 + 3x2. Find the zeros of P(x) and sketch its graph.

  10. Shape of the Graph Near a Zero: If the factor (x − c) appears m times in the complete factorization of P(x), i.e., P(x) = (x − c)m · Q(x) with Q(c) ≠ 0, then c is said to be a zero of multiplicity m. If c is a zero of even multiplicity, then the graph of y = P(x) touches the x-axis at (c, 0). If c is a zero of odd multiplicity, then the graph of y = P(x) crosses the x-axis at (c, 0).

  11. Example 5: Let f(x) = x3(x2 + 1)(x − 3)4. List each real zero and its multiplicity. Determine if the graph crosses or touches the x-axis. Find the degree and the end behavior of the function. Find the y-intercept of y = f(x). And x = 3 is a zero of multiplicity 4 Not Possible! Consequently, x=0 is a zero of multiplicity 3

  12. Since x = 0 has multiplicity 3 which is odd the graph crosses the axis Since x = 3 has multiplicity 4 which is even the graph touches the axis ? So y → +∞ as x → +∞ and y → −∞ as x → −∞

  13. Since x = 0 has multiplicity 3 which is odd the graph crosses the axis Since x = 3 has multiplicity 4 which is even the graph touches the axis

  14. Local Maxima and Minima of Polynomials: If a point (a, f(a)) is the highest point on the graph of f within some viewing rectangle, then f(a) is a local maximum value of f. If a point (b, f(b)) is the lowest point on the graph of f within some viewing rectangle, then f(b) is a local minimum value of f.

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