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4 Real Zeros of Polynomial Functions.notebook 1 August 07, 2012 Real Zeros of Polynomial Functions
Transcript of 4 Real Zeros of Polynomial Functions.notebook · 4 Real Zeros of Polynomial Functions.notebook 9...
4 Real Zeros of Polynomial Functions.notebook
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Real Zeros of Polynomial Functions
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Real Zeros of Polynomial Functions
The Division Algorithm:
• f(x) and d(x) are polynomials d(x) ≠0
• Degree of d(x) ≤ f(x)
• If r(x)=0, then d(x) divides evenly into f(x).
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Long Division. . .
Are you smarter than a fifth grader?
EX#1: Let’s look at this rule with simple numbers.
Find the quotient, remainder, dividend and divisor of 763÷3
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Long Division vs. Synthetic Division
EX#2: Divide using Long Division
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EX#3: Divide using Synthetic Division
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THE REMAINDER AND FACTOR THEOREMS
If a polynomial f(x) is divided by x – k , the remainder is r = f(k)
EX#4: Use the remainder theorem to evaluatewhen
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EX#5: Is a factor of?
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EX#6: Verify that and are factors of ind all the zeros of f.
Using The Remainder Theorem
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2. If, k is a factor of .
3. k an
Uses of the Remainder in Synthetic Division
The remainder r, obtained in synthetic division ofby provides the following:
1. The remainde r gives the value of at k
That is,
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The Rational Zeros Theorem
If is a polynomial of degree indescending order then it is possible to make an organized list of potential rational numbers to aid in inding zeros of the function.
Possible Rational = Zeros
EX#7: Find the rational zeros of:
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More Rational Zeros and Real Zeros
EX#8: Find the rational zeros of:
Ex#9: Find all the real zeros of
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Modeling
EX#10: Find a polynomial function with leading coeficient 2 that has the given degree and zeros. Degree3,with zerosat
EX.#11: Find the cubic function with the given table of values. Check with a grapher.
x –2 1 2 3y 0 24 0 0
