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

,...)( 011

1 axaxaxaxP nn

nn

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.

End Behavior and the Leading Term:

0na 0na

0na 0na

Real Zeros of Polynomials:

If P(x) is a polynomial and c is a real number, thenthe following are equivalent:1.c is a zero of P(x);2.P(c) = 0;3.x − c is a factor of P(x);4.(c,0) is an x-intercept of the graph of y = P(x).

Example 1:

Determine the end behavior of thepolynomial 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 → −∞

32 143 x - - x P(x) slower term43 32 x - x

slower term123 32 xx

slower term3 5 x

?

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.

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].

12223)2()2( 34 P 148*316 142416 148

14 5

13233)3()3( 34 P 168181 16 5

Since P(2)>0 and P(3)< 0 there must be some number c between 2 and 3 such that P(c) = 0.

Example 3:

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

:are zeros The

2x 1x

3x3 is termleading The x

6,0

Example 4:

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

234 32)( xxxxP

42 is termleading The x

32)( 22 xxxxP

132)( 2 xxxxP

:are zeros The20 x 10 x 320 x

x0 x23

x2

3x1

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 ofy = P(x) crosses the x-axis at (c, 0).

Example 5:

Let f(x) = x3(x2 + 1)(x − 3)4.1.List each real zero and its multiplicity.2.Determine if the graph crosses or touches the x-axis.3.Find the degree and the end behavior of the function.4.Find the y-intercept of y = f(x).

whenoccure zeros The

03) -1)(x (xx 423

0x 3 01) (x2 03) -(x 4 0x 1 x2

Not Possible!

03) -(x 3x

Consequently, x=0 is a zero of multiplicity 3

And x = 3 is a zero of multiplicity 4

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

423 3) -1)(x (xxf(x)

slower term1 423 x xx

slower term435 x x x

slower term9 x

So y → +∞ as x → +∞and y → −∞ as x → −∞

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intercept-y

0x 423 301000 f

0 0,0

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

slower term9 x423 3) -1)(x (xxf(x)

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.