What is the value of $\dfrac{f(-5)+f(9)}{4}$ in the Polynomial Challenge III?

In summary, we are given a function $f(x)$ with real constants and three specific values for $f(x)$. We are asked to find the value of a new expression involving $f(x)$.
  • #1
anemone
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Let $f(x)=x^4+px^3+qx^2+rx+s$, where $p,\,q,\,r,\,s$ are real constants. Suppose $f(3)=2481$, $f(2)=1654$, $f(1)=827$.

Determine the value of $\dfrac{f(-5)+f(9)}{4}$.
 
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  • #2
anemone said:
Let $f(x)=x^4+px^3+qx^2+rx+s$, where $p,\,q,\,r,\,s$ are real constants. Suppose $f(3)=2481$, $f(2)=1654$, $f(1)=827$.

Determine the value of $\dfrac{f(-5)+f(9)}{4}$.

Hello.

[tex]f(1)=827=827*1[/tex]

[tex]f(2)=1654=827*2[/tex]

[tex]f(3)=2481=827*3[/tex]

Therefore:

[tex]\dfrac{f(-5)+f(9)}{4}=\dfrac{827*(-5)+827*(9)}{4}=\dfrac{827(-5+9)}{4}=827[/tex]

Regards
 
  • #3
mente oscura said:
Hello.

[tex]f(1)=827=827*1[/tex]

[tex]f(2)=1654=827*2[/tex]

[tex]f(3)=2481=827*3[/tex]

Therefore:

[tex]\dfrac{f(-5)+f(9)}{4}=\dfrac{827*(-5)+827*(9)}{4}=\dfrac{827(-5+9)}{4}=827[/tex]

Regards

Good try! But that isn't correct...sorry!:)
 
  • #4
We have f(1) = 827 *1, f(2) = 827*2 , f(3) = 827*3

Hence f(x) – 827 x = 0 for x = 1, 2, or 3

So f(x) = a(x-b)(x-1)(x-2)(x-3) + 827 x where b is the 4th zero of f(x)- 827 x

Comparing coefficient of $x^4$ we have a = 1

so f(x) = (x-b)(x-1)(x-2)(x-3) + 827 x

rest is as below

f(9) = (9-b) * 8 * 7 * 6 + 827 * 9 = 336(9-b) + 827 * 9

f(-5) = (-5-b) * (-6) *(-7) * (-8) + 827 * (-5) = 336(5+b) + 827 * (-5)

So (f(9) + f(-5))/4 = (336 * 14 + 827 * (4))/4 = 2003
 
  • #5
kaliprasad said:
We have f(1) = 827 *1, f(2) = 827*2 , f(3) = 827*3

Hence f(x) – 827 x = 0 for x = 1, 2, or 3

So f(x) = a(x-b)(x-1)(x-2)(x-3) + 827 x where b is the 4th zero of f(x)- 827 x

Comparing coefficient of $x^4$ we have a = 1

so f(x) = (x-b)(x-1)(x-2)(x-3) + 827 x

rest is as below

f(9) = (9-b) * 8 * 7 * 6 + 827 * 9 = 336(9-b) + 827 * 9

f(-5) = (-5-b) * (-6) *(-7) * (-8) + 827 * (-5) = 336(5+b) + 827 * (-5)

So (f(9) + f(-5))/4 = (336 * 14 + 827 * (4))/4 = 2003

Well done, kaliprasad and thanks for participating!
 
  • #6
anemone said:
Let $f(x)=x^4+px^3+qx^2+rx+s$, where $p,\,q,\,r,\,s$ are real constants. Suppose $f(3)=2481$, $f(2)=1654$, $f(1)=827$.

Determine the value of $\dfrac{f(-5)+f(9)}{4}$.

Define $g(x)=f(x+2)$. Then:
$$g(x)=x^4+Px^3+Qx^2+Rx+S$$
for some real constants $P, Q, R, S$.It follows that:
$$g(0) = S = 1654$$
$$g(1)+g(-1)=(1 + P + Q + R + S) + (1-P+Q-R+S) = 2+2Q+2S = 2481 +827 = 3308$$
$$Q = \frac{3308 - 2 - 2S}{2} = 1653 - S = 1653 - 1654 = -1$$

$$\dfrac{f(-5)+f(9)}{4} = \dfrac{g(-7)+g(7)}{4}
=\frac 1 4 (2\cdot 7^4 + 2 Q\cdot 7^2 + 2S)
=\frac 1 4 (2\cdot 7^4 - 2 \cdot 7^2 + 2\cdot 1654)
=2003$$
 
  • #7
I like Serena said:
Define $g(x)=f(x+2)$. Then:
$$g(x)=x^4+Px^3+Qx^2+Rx+S$$
for some real constants $P, Q, R, S$.It follows that:
$$g(0) = S = 1654$$
$$g(1)+g(-1)=(1 + P + Q + R + S) + (1-P+Q-R+S) = 2+2Q+2S = 2481 +827 = 3308$$
$$Q = \frac{3308 - 2 - 2S}{2} = 1653 - S = 1653 - 1654 = -1$$

$$\dfrac{f(-5)+f(9)}{4} = \dfrac{g(-7)+g(7)}{4}
=\frac 1 4 (2\cdot 7^4 + 2 Q\cdot 7^2 + 2S)
=\frac 1 4 (2\cdot 7^4 - 2 \cdot 7^2 + 2\cdot 1654)
=2003$$

Bravo, I like Serena! Hey, your method gives me a very good insight on how to define for another function to relate to the given function and use it to our advantage! Thanks for that and thanks for participating.:)
 

FAQ: What is the value of $\dfrac{f(-5)+f(9)}{4}$ in the Polynomial Challenge III?

What is the purpose of "Polynomial Challenge III"?

The purpose of "Polynomial Challenge III" is to test participants' knowledge and skills in solving complex polynomial equations and functions.

How difficult is "Polynomial Challenge III" compared to the previous challenges?

"Polynomial Challenge III" is significantly more difficult and challenging compared to the previous challenges, as it involves more advanced concepts and techniques in solving polynomial equations.

Can I use a graphing calculator for "Polynomial Challenge III"?

No, the use of calculators or any other external aids is not allowed for "Polynomial Challenge III". Participants are expected to solve the equations and functions manually.

Are there any specific strategies or techniques that can help in solving "Polynomial Challenge III"?

Yes, there are various strategies and techniques that can be used to solve "Polynomial Challenge III", such as factoring, synthetic division, and the rational root theorem. It is recommended to practice and familiarize oneself with these techniques before attempting the challenge.

Is there a time limit for completing "Polynomial Challenge III"?

Yes, there is a time limit of 60 minutes for completing "Polynomial Challenge III". Participants must complete and submit their solutions within this time frame to be considered for the challenge.

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