Homework “Mini-Quiz”Homework “Mini-Quiz”Use the paper provided - 10 min. (NO TALKING!!)Use the paper provided - 10 min. (NO TALKING!!)

Do NOT write the question – Answer Only!!Do NOT write the question – Answer Only!!1) A polynomial function written with terms in descending degree is

written in _______ form. 2) In the function above, n is called ______________.3) Describe how f(x) = -(x+2)3 - 1 would change g(x) = x3.4) Given any polynomial function f(x) = anxn+..+a1x+a0, if an > 0 and

n is ____, then and

5) Given any polynomial function f(x) = anxn+..+a1x+a0, if an ___ 0 and n is even, then and

6) Graph f(x) = (x-2)2(x+1)(x-3). Describe the end behavior using limits. (#21)

7) Find the zeros of f(x) = x3-25x algebraically. (Show your work) (#36)

If you finish before the timer sounds, start the warm-up. (NO TALKING!!)

Warm-up (5 min.)Warm-up (5 min.)

1) Using only algebra, find a cubic function with the given zeros: 3,-4,6

2) Use cubic regression to fit a curve through the four points given in the table:

x -2 1 4 7

y 2 5 9 26

2.4 Real Zeros of Polynomial Functions2.4 Real Zeros of Polynomial Functions

• Divide polynomials using long division or synthetic division

• Apply the Remainder and Factor Theorem• Find upper and lower bounds for zeros of

polynomials

Do you recall?Do you recall?

In the long division shown, what are the names for the values: 30, 4, 7, and 2?

7

2

28

304

Ex1 Using long division to divide f(x) by d(x), and Ex1 Using long division to divide f(x) by d(x), and write a summary statement in polynomial form and write a summary statement in polynomial form and

fraction form.fraction form.

a) f(x) = x2 - 2x + 3; d(x) = x – 1

b) f(x) = x4 - 2x3 + 3x2 - 4x + 6; d(x) = x2 + 2x - 1

Long Division and the Division Long Division and the Division AlgorithmAlgorithm

Let f(x) and d(x) be polynomials with the degree of f greater than or equal to the degree of d, and d(x) 0. Then there are unique polynomials q(x) and r(x), called the quotient and remainder, such that

f(x) = d(x) q(x) + r(x)

where either r(x) = 0 or the degree of r is less than the degree of d.

The remainder determines a factorThe remainder determines a factor

• Remainder Theorem - If a polynomial f(x) is divided by x - k, then the remainder is r = f(k).

• Factor Theorem - A polynomial function f(x) has a factor x - k if an only if f(k) = 0.

Ex 2 Use the remainder theorem to find the Ex 2 Use the remainder theorem to find the remainder when f(x) = 2xremainder when f(x) = 2x22 - 3x + 1 is divided by: - 3x + 1 is divided by:

a) x – 2

b) x + 4

Fundamental Connections for Polynomial FunctionsFundamental Connections for Polynomial Functions

For a polynomial function f and a real number k, the following statements are equivalent:

1. x = k is a solution (or root) of the equation f(x) = 0.

2. k is a zero of the function f.

3. k is an x-intercept of the graph of y = f(x).

4. x - k is a factor of f(x).

Ex 3 Use synthetic division to divide Ex 3 Use synthetic division to divide f(x) = xf(x) = x33 + 5x + 5x22 + 3x – 2 by: + 3x – 2 by:

a) x + 1

b) x - 2

Synthetic DivisionSynthetic Division1. Express the polynomial in standard form.2. Use the coefficient (including zero coefficients) for

synthetic division.3. Find the zero of the divisor (x - k = 0)4. Use this as the divisor in the synthetic division5. Bring down the leading coefficient6. Multiply by the “zero divisor”7. Add this product to the next coefficient8. Repeat steps 4 & 5 until all coefficients have been

used9. The last coefficient is the remainder10. The other coefficient are the coefficients for the

quotient polynomial when written in standard form.

Upper and Lower BoundsUpper and Lower BoundsLet f be a polynomial function of degree n > 1 with a positive leading coefficient. Suppose f(x) is divided by x – k using synthetic division.

• If k > 0 and every number in the last line is nonnegative (positive or zero), then k is an upper bound for the real zeros of f.

• If k < 0 and the numbers in the last line are alternately nonnegative and nonpositive, then k is a lower bound for the real zeros of f.

Ex 4 Use synthetic division to prove that the number k Ex 4 Use synthetic division to prove that the number k is the upper or lower bound (as stated) for the real is the upper or lower bound (as stated) for the real

zeros of the function f.zeros of the function f.

a) k = 3 is an upper bound; f(x) = 2x3 – 4x2 + x – 2

b) k = -1 is a lower bound; f(x) = 3x3 – 4x2 + x + 3

Searching for zerosSearching for zeros

Ex 5 Show that all the zeros of f(x) = 2x3 – 3x2 – 4x + 6 lie within the interval [-7,7]. Find all of the zeros.

Rational Zeros (Roots) TheoremRational Zeros (Roots) Theorem

If a polynomial f(x) = anxn + an-1xn-1 + … + a1x + a0 has any rational roots, then they are of the form p/q where q is a factor of an (the leading coefficient) and p is a factor of a0 (the constant term).

Ex 6 List all the possible rational roots of

f(x) = 2x3 + 5x2 - 3x + 5

Tonight’s AssignmentTonight’s Assignment

p. 216 - 218 Ex 3-33 m. of 3, 39, 42, 51-60 m. of 3

Exit TicketExit Ticket

• Find all the roots of f(x) from Ex 6 and place in the turn in box before you leave.

• Have a great day!!

• Remember to study!!