MHB Evaluating $f(15)$ without a Calculator

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The discussion focuses on evaluating the polynomial function f(x) at x=15 without a calculator. Participants share methods for simplifying the expression, highlighting the structure of the polynomial and its coefficients. Techniques such as factoring and substitution are discussed to arrive at the value of f(15). The conversation emphasizes collaborative problem-solving and encourages engagement among participants. Ultimately, the evaluation of f(15) showcases the effectiveness of analytical approaches in polynomial calculations.
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Let $f(x)= x^{11}-16x^{10}+16x^9-16x^8+16x^7-16x^6+16x^5-16x^4+16x^3-16x^2+16x-1$.

Evaluate $f(15)$ without using the calculator.
 
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anemone said:
Let $f(x)= x^{11}-16x^{10}+16x^9-16x^8+16x^7-16x^6+16x^5-16x^4+16x^3-16x^2+16x-1$.

Evaluate $f(15)$ without using the calculator.

we have

$f(x) = x^{10}(x-15) - x^9(x-15) + x^8(x-15) - x^7(x-15) + x^6(x-15) - x^5(x-15) + x^4(x-15) - x^3(x-15) + x^2(x-15) - x(x-15) + x - 1$

so f(15) = 14
 
kaliprasad said:
we have

$f(x) = x^{10}(x-15) - x^9(x-15) + x^8(x-15) - x^7(x-15) + x^6(x-15) - x^5(x-15) + x^4(x-15) - x^3(x-15) + x^2(x-15) - x(x-15) + x - 1$

so f(15) = 14

Very well done, kaliprasad, and thanks for participating!:)
 
By inspection, $$x-1$$ is a factor of $$f(x)$$. After polynomial long division, we have

$$f(x)=(x-1)(x^{10}-15x^9+x^8-15x^7+x^6-15x^5+x^4-15x^3+x^2-15x+1)$$

so

$$f(15)=14\cdot1=14$$
 
Last edited:
greg1313 said:
By inspection, $$x-1$$ is a factor of $$f(x)$$. After polynomial long division, we have

$$f(x)=(x-1)(x^{10}-15x^9+x^8-15x^7+x^6-15x^5+x^4-15x^3+x^2-15x+1)$$

so

$$f(15)=14\cdot1=14$$

Very good job, greg1313! And thanks for participating! :)
 
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