Evaluating $f(15)$ without a Calculator

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Discussion Overview

The discussion revolves around 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 the point $x=15$ without the use of a calculator. The focus is on exploring methods or reasoning to arrive at the value of $f(15)$.

Discussion Character

  • Exploratory, Homework-related, Mathematical reasoning

Main Points Raised

  • Post 1 presents the function and the task of evaluating it at $x=15$.
  • Post 2 reiterates the function and the evaluation task, suggesting a focus on the problem itself.
  • Post 3 acknowledges a participant's contribution positively, indicating engagement but not addressing the evaluation directly.
  • Post 4 similarly praises another participant's effort without providing further mathematical insight.

Areas of Agreement / Disagreement

There is no clear agreement or disagreement on the evaluation method, as the discussion primarily consists of participants acknowledging each other's contributions without delving into the mathematical evaluation itself.

Contextual Notes

The posts do not provide any mathematical reasoning or steps toward evaluating $f(15)$, leaving the discussion open-ended regarding the approach to take.

Who May Find This Useful

This discussion may be of interest to those looking for collaborative problem-solving approaches in polynomial evaluation or participants seeking to engage with others in mathematical challenges.

anemone
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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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