Question 124200: 2.Suppose that the polynomial f(x) is given f(x) =a.x^2+b.x+c.Suppose ,further,that f(0)=1, f(1)=2, and f(a)=12. Find a, b and c.
Please help me.help..
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! Since nobody else has responded to this, here's a method you can use, but the solution is
not a trivial exercise.
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You are given 
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Finding c is easy because f(0) = 1. Substitute 0 for x and the given equation becomes:
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and since f(0) = 1 you can say that the right side of this equation equals 1. So:
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and this reduces to  . That's one answer. You can now replace c with 1 in the given
equation to get:
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Next you are told that f(1) = 2. That means that in:
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when you replace x with 1 you get:
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This reduces to:
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Solve this for b by adding -a - 1 to both sides to get:
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Combining the two constants (2 - 1) on the right side reduces the equation to:
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Remember this result. Next, in the equation:
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make use of f(a) = 12. If you substitute a for x the equation becomes:
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but recall that . Therefore, replace b with  and you get:
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Do the multiplication of the terms in a and you have:
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Subtract 12 from both sides and rearrange in descending powers of a. When you do those two
things you get:
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You can now solve this equation for a, but the answer you get is not "nice." I question
whether you copied f(0), f(1), or f(a) correctly since a mistake there would impact the
answer. However, moving on, I used a calculator to solve this cubic equation for a and got:
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and recall from above that . So, substituting 2.43905659 for a results in:
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So, we now have as answers: a = 2.43905659 and b = -1.43905659 and c = 1.
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To check, we substitute these into the original given equation of:
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and this equation becomes:
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Check that f(0) = 1 by substituting 0 for x to get:
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That checks.
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Next, check that f(1) = 2 by substuting 1 for x to get:
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That checks.
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Next, check that f(a) = 12 by substituting a for x ... that is to say, substituting
2.43905659 for x to see if the result is 12. Substituting 2.43905659 for x gives you:
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Doing the multiplication of (2.43905659)(2.43905659^2) and of (1.43905659)(2.43905659) reduces
the equation to:
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and that checks too.
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So the values found for a, b, and c are correct. The values for a and b are not nice, but
they work and are therefore correct.
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Hope this helps ...
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