Polynomials Question 1  9&10 Math Capsule  Misbah Sir  Infinity Learn Class 9&10  Summary and Q&A
TL;DR
Given a polynomial and its zeros, find the value for a given number by using the concept of polynomial zeros.
Key Insights
 😥 The value of a polynomial at certain points can be found by substituting those points into the polynomial representation formed by its zeros.
 0️⃣ The concept of zeros of a polynomial helps represent the polynomial accurately and simplify calculations.
 😥 Using the concept of polynomial zeros, finding the value of P(4) + P(0) can be done more efficiently than individually substituting each point.
Transcript
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Questions & Answers
Q: How can you find the value of P(4) and P(0) in a given polynomial with given zeros?
To find P(4), we can substitute 4 into the polynomial representation using the zero factors. Similarly, to find P(0), we substitute 0 into the polynomial representation.
Q: Why can we assume one more zero, α, for the polynomial with given zeros?
Since the polynomial has a degree of four and three zeros are given, according to the fundamental theorem of algebra, there must be one more zero, represented by α.
Q: Why can we represent the polynomial as (x  1)(x  2)(x  3)(x  α)?
Using the concept of polynomial zeros, we know that each zero corresponds to a factor in the polynomial. Therefore, by multiplying the factors corresponding to the given zeros, we can represent the polynomial accurately.
Q: How do we simplify the expression for P(4) and P(0) using the polynomial representation?
Substituting 4 into the polynomial representation gives us 24  6α for P(4), and substituting 0 gives us 6α for P(0). Therefore, P(4) + P(0) simplifies to 24.
Summary & Key Takeaways

The question asks to find the value of P(4) + P(0) for a polynomial with given zeros.

By using the concept of polynomial zeros and representing the polynomial with its factors, the question can be solved more efficiently.

The value of P(4) is found to be 24  6α, and the value of P(0) is found to be 6α. Therefore, P(4) + P(0) equals 24.