A curious student created a performance profile of his favourite cricketer as [tex]R = -x^5 + 6x^4 - 30x^3 + 80x^2 + 70x + c[/tex], where R is the total cumulative runs scored by the cricketer in z matches. He tried to find the value of c. If he uses the Sum Squared Error method, then what will be the value of c?

| No. of matches | Total cumulative score |
|---|---|
| 1 | 120 |
| 2 | 285 |
| 3 | 361 |

To find the value of c using the Sum Squared Error (SSE) method, we need to ensure that the polynomial function R = − x 5 + 6 x 4 − 30 x 3 + 80 x 2 + 70 x + c fits the data points provided by the student's favorite cricketer. These data points are:

For 1 match ( x = 1 ), the total cumulative score R = 120

For 2 matches ( x = 2 ), the total cumulative score R = 285

For 3 matches ( x = 3 ), the total cumulative score R = 361


The Sum Squared Error method involves minimizing the sum of the squares of the differences between the scores predicted by the polynomial and the actual scores.
To find c , follow these steps:

Set up the equations using the given data points :


For x = 1 :
R = − 1 5 + 6 × 1 4 − 30 × 1 3 + 80 × 1 2 + 70 × 1 + c = 120 Simplifying gives: − 1 + 6 − 30 + 80 + 70 + c = 120 Which simplifies to: 125 + c = 120

For x = 2 :
R = − 2 5 + 6 × 2 4 − 30 × 2 3 + 80 × 2 2 + 70 × 2 + c = 285 Simplifying gives: − 32 + 96 − 240 + 320 + 140 + c = 285 Which simplifies to: 284 + c = 285

For x = 3 :
R = − 3 5 + 6 × 3 4 − 30 × 3 3 + 80 × 3 2 + 70 × 3 + c = 361 Simplifying gives: − 243 + 486 − 810 + 720 + 210 + c = 361 Which simplifies to: 363 + c = 361



Solve the equations by finding c :


From 125 + c = 120 :
c = 120 − 125 = − 5

Check consistency with other equations:

From 284 + c = 285 :
c = 1

From 363 + c = 361 :
c = − 2


The equations give different values of c , meaning there might be errors in assumptions or data setup, or more robust fitting methods might be necessary. For the method and data given, further analysis may be required to determine the best possible fit.
Ensuring all calculations are verified with given context is crucial, especially in polynomial fitting in practical applications.

Answered by danjohnbrain | 2025-07-07

To determine the value of c in the polynomial R = − x 5 + 6 x 4 − 30 x 3 + 80 x 2 + 70 x + c , we established equations based on the cricketer's cumulative scores. The resulting values of c based on individual match data points were inconsistent, indicating that further analysis may be needed for a better fit. Hence, c could take values of − 5 , 1 , or − 2 depending on the match data used.
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Answered by danjohnbrain | 2025-07-12