Evaluating $f(15)$ without a Calculator

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The discussion focuses on evaluating the polynomial function $f(x) = x^{11} - 16x^{10} + 16x^9 - 16x^8 + 16x^7 - 16x^6 + 16x^5 - 16x^4 + 16x^3 - 16x^2 + 16x - 1$ at $x=15$ without using a calculator. Participants successfully engage in the evaluation process, demonstrating effective strategies for handling high-degree polynomials. The contributions from users kaliprasad and greg1313 highlight collaborative problem-solving and provide insights into polynomial evaluation techniques.

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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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