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Published byAmos Green Modified over 9 years ago
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FINDING A POLYNOMIAL PASSING THROUGH A POINT
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Review: the Linear Factorization Theorem If where n > 1 and a n ≠ 0 then Where c 1, c 2, … c n are complex numbers (possibly real and not necessarily distinct) This theorem lets us generate polynomials with any zeroes by multiplying their corresponding factors.
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Specific Points If we want to find a polynomial with given zeroes that passes through a point, we can use basically the same technique. We’ll still multiply together all the factors that correspond to the zeros of the polynomial, but we’ll also multiply by a constant term (a n ) that results in the point we want.
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The Technique 1. Determine all the zeroes you want your polynomial to have and what multiplicity each should have. 2. Generate a factor for each zero. 3. Multiply together all the factors. Multiply by each one a number of times equal to its multiplicity. 4. Plug in the point that you want the polynomial to pass through and determine the value of a n.
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Example Find a third degree polynomial with zeroes -1, 1, and 3 that passes through the point (2, -6).
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Solution: Step 1 The first step is to determine all the zeroes and their multiplicities. With three zeros, we already have enough factors to form a third degree polynomial, so all of our zeros will have multiplicity 1. The zeroes, again, are -1, 1, and 3.
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Solution: Step 2 Since our zeroes are -1, 1, and 3, our three factors are (x + 1), (x – 1), and (x – 3).
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Solution: Step 3 Now, we multiply all the factors together and expand the product. This gives us x 3 – 3x 2 – x + 3. Our equation, in its current form, is f(x) = a n (x 3 – 3x 2 – x + 3). Now we just need to find a n. If we just wanted a polynomial with the given zeroes, we could choose any value of a n (usually 1), but we need our function to go through a particular point.
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Solution: Step 4 To find a n, we plug in our point – (2, -6). This gives us -6= a n (2 3 – 3*2 2 – 2 + 3). -6 = a n (8 – 12 – 2 + 3) -6 = -3a n a n = 2 Our final polynomial is f(x) = 2x 3 – 6x 2 – 2x + 6.
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Graph We can check our answer by checking whether the graph of the function goes through the point we wanted. It does!
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