Evaluating $f(15)$ without a Calculator

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In summary, evaluating $f(15)$ without a calculator involves using mathematical techniques such as substitution, simplification, and algebraic manipulation. It is important because it improves problem-solving skills and allows for a deeper understanding of mathematical concepts. Common techniques include using the order of operations, factoring, and trigonometric identities. An example of evaluating $f(15)$ without a calculator is substituting the value of x into the given function and simplifying the equation. Tips for making it easier include breaking down the problem into smaller parts and using common mathematical properties and formulas.
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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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  • #2
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
 
  • #3
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!:)
 
  • #4
By inspection, \(\displaystyle x-1\) is a factor of \(\displaystyle f(x)\). After polynomial long division, we have

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

so

\(\displaystyle f(15)=14\cdot1=14\)
 
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  • #5
greg1313 said:
By inspection, \(\displaystyle x-1\) is a factor of \(\displaystyle f(x)\). After polynomial long division, we have

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

so

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

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

FAQ: Evaluating $f(15)$ without a Calculator

How do you evaluate $f(15)$ without a calculator?

Evaluating $f(15)$ without a calculator involves using mathematical techniques and concepts such as substitution, simplification, and algebraic manipulation to solve the given equation or function.

Why is it important to be able to evaluate $f(15)$ without a calculator?

Being able to evaluate $f(15)$ without a calculator allows for a deeper understanding of mathematical concepts and improves problem-solving skills. It also helps in situations where a calculator is not available or accurate enough.

What are some common techniques used to evaluate $f(15)$ without a calculator?

Some common techniques include using the order of operations, factoring, and using trigonometric identities. Other methods may include using logarithms, the quadratic formula, or the binomial theorem.

Can you provide an example of how to evaluate $f(15)$ without a calculator?

Sure, let's say we have the function $f(x) = 3x^2 + 5x - 2$, and we want to evaluate $f(15)$. We would first substitute 15 for x in the function, giving us: $f(15) = 3(15)^2 + 5(15) - 2$. Then, we would simplify the equation to get the final answer of $f(15) = 698$.

Are there any tips for making it easier to evaluate $f(15)$ without a calculator?

One tip is to break down the problem into smaller parts and simplify as much as possible before substituting the value of x. It can also be helpful to know and use common mathematical properties and formulas, such as the distributive property, to simplify the equation.

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