# chapter 11 polynomial functions 11.1 polynomials and polynomial functions

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- Chapter 11 Polynomial Functions 11.1 Polynomials and Polynomial Functions
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- Chapter 11 Polynomial Functions 11.1 Polynomials and Polynomial Functions
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- A polynomial function is a function of the form f (x) = a n x n + a n 1 x n 1 + + a 1 x + a 0 Where a n 0 and the exponents are all whole numbers. A polynomial function is in standard form if its terms are written in descending order of exponents from left to right. For this polynomial function, a n is the leading coefficient, a 0 is the constant term, and n is the degree. a n 0 anan anan leading coefficient a 0a 0 a0a0 constant term n n degree descending order of exponents from left to right. n n 1
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- Objective: Determine whether a number is a root or zero of a given equation or function.
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- Objective: Determine whether one polynomial is a factor of another by division.
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- HW #11.1 Pg 483-484 1-21 Odd, 22-31, 35-36
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- Chapter 11 Polynomial Functions 11.2 Factor and Remainder Theorems
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- P(10) is the remainder when P(x) is divided by x - 10.
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- P(10) = 73,120 P(-8) = -37, 292
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- Find P( -4)
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- Yes No Yes
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- We look for linear factors of the form x - r. Let us try x - 1. We know that x - 1 is not a factor of P(x). We try x + 1. To solve the equation P(x) = 0, we use the principle of zero products.
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- P(x) = (x 2)(x + 3)(x + 5) x = 2 x = -3 x = -5
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- 4. Solve -5 < x< 1 or 2 < x < 3
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- HW #11.2 Pg 488-489 1-15 Odd, 16-31
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- Chapter 11 11.3 Theorems about Roots
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- Carl Friedrich Gauss was one of the great mathematicians of all time. He contributed to many branches of mathematics and science, including non-Euclidean geometry and curvature of surfaces (later used in Einstein's theory of relativity). In 1798, at the age of 20, Gauss proved the fundamental theorem of algebra.
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- If a factor (x - r) occurs k times, we say that r is a root of multiplicity k Where in the ____ did that come from? The polynomial has 5 linear factors and 5 roots. The root 2 occurs 3 times, however, so we say that the root 2 has a multiplicity of 3.
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- -7 Multiplicity 2 3 Multiplicity 1 4 Multiplicity 2 3 Multiplicity 2 1 Multiplicity 1 -1 Multiplicity 1
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- Degree 3 3 roots Complex Roots Occur in Conjugate Pairs
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- Irrational Roots also come in Conjugate Pairs Degree 6 6 roots
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- Degree 4 4 roots 2i -2i 1.Divide p(x) by a known root to reduce it to a polynomial of lesser degree 2.Divide the result by a different known root to reduce the degree again 3.Repeat Steps 1 and 2 until you have reduced it to degree 2, then factor or use the quadratic formula to find the remaining roots Roots are 2i, -2i, 2, and 3.
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- The number a n can be any nonzero number. Let a n = 1.
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- We proceed as in Example 6, letting a n = 1 Degree 5 5 roots Multiplicity 3 means it is a factor 3 times
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- HW #11.3 Pg 494-495 1-49 Odd, 59
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- 4 3 No
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- Chapter 11 11.4 Rational Roots
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- List the possible rational zeros.
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- Test these zeros using synthetic division. The roots of are -1, 3, and -4.
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- List the possible rational zeros. Test these zeros using synthetic division.
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- The roots of are -2,, and.
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- x = 1 x = -1
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- HW # 11.4 Pg 499-500 1-11Odd, 13-21, 23-27 Odd
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- Chapter 11 11-5 Descartes Rule of Signs
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- Theorem 11-8 Descartes Rule Of Signs Part #1 The number of positive real zeros of a polynomial P(x) with real coefficients is a.the same as the number of variations of the sign of P(x), or b.Less than the number of variations of sign of P(x) by a positive even integer starts Pos. changes Neg. changes Pos. 12 There are 2 sign changes so this means there could be 2 or 0 positive real zeros to the polynomial.
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- Determine the number of positive real zeros of the function EXAMPLES 1 +-++ 2 Sign Changes 2 or 0 Positive Real Roots 2 +-+- 4 Sign Changes 4, 2, or 0 Positive Real Roots +
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- Determine the number of positive real zeros of the function EXAMPLES 3 +-- 1 Sign Changes Exactly 1 Positive Real Roots
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- Try ThisDetermine the number of positive real zeros of the function.
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- Theorem 11-8 Descartes Rule Of Signs Part #2 The number of negative real zeros of a polynomial P(x) with real coefficients is a.the same as the number of variations of the sign of P(-x), or b.Less than the number of variations of sign of P(-x) by a positive even integer There are 2 sign changes so this means there could be 2 or 0 negative real zeros to the polynomial. starts Pos. changes Neg. changes Pos. 12
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- Determine the number of negative real zeros of the function EXAMPLES 4 +-+ 4 Sign Changes 4, 2, or 0 Negative Real Roots -+
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- Try ThisDetermine the number of negative real zeros of the function.
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- 686769
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- If a sixth-degree polynomial with real coefficients has exactly five distinct real roots, what can be said of one of its roots? Is it possible for a cubic function to have more than three real zeros? Is it possible for a cubic function with real coefficients to have no real zeros?
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- HW #11.5 Pg 503 1-32
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- Chapter 11 11-6 Graphs of Polynomial Functions
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- 3. 4. 5.
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- First, plot the x-intercepts. Second, use a sign chart to determine when f(x) > 0 and f(x) < 0 3 0 0 + ++ + + + f(0) =3, Sketch a smooth curve + + +
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- First, plot the x-intercepts. Second, use a sign chart to determine when f(x) > 0 and f(x) < 0 -2 1 0 0 + - + + + + f(0) =2, Sketch a Smooth Curve + - +
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- First, plot the x-intercepts. Second, use a sign chart to determine when f(x) > 0 and f(x) < 0 -2 0 0 + - - + - + f(0) =-12, Sketch a Smooth Curve + - - 3 (0, -12) 0 + + + + +- +
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- A B
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- 3 x-intercepts 3 real roots. 1 x-intercept, 1 real root 2 x-intercepts, 2 real roots. The left and right ends of a graph of an odd-degree function go in opposite directions.
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- 4 x-intercepts 4 real roots. 1 x-intercept, 1 real root 2 x-intercepts, 2 real roots. The left and right ends of a graph of an even-degree function go in the same directions. 3 x-intercepts, 3 real roots.
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- Even Multiplicity Odd Multiplicity
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- 3. Factor and make a sign chart. 5. Plot this information and consider the sign chart.
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- HW #11.6 Pg 507-508 1-22
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- Test Review
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- 12
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- 4. Solve -5 < x< 1 or 2 < x < 3
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- The coefficient of x n-1 is the negative of the sum of the zeros.
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- HW #R-11a Pg 511-512 1-22
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