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Which of the following polynomials lies in the linear span of S = \{1, 1 + x + x ^ 2\} ? (a) 5x ^ 2 + 5x + 1 (b) 100x ^ 2 + 10x + 10 (c) pi(x ^ 2 + 1) (d) 7x ^ 2 + pi*x + 3?
Most Upvoted Answer
Which of the following polynomials lies in the linear span of S = \{1,...
Understanding the Linear Span
To determine which polynomial lies in the linear span of \( S = \{1, 1 + x + x^2\} \), we need to check if each polynomial can be expressed as a linear combination of the elements in \( S \).
Elements of S
- The first element is \( 1 \).
- The second element is \( 1 + x + x^2 \).
Formulating the Linear Combination
A polynomial \( P(x) \) lies in the linear span of \( S \) if it can be expressed as:
\[
P(x) = a(1) + b(1 + x + x^2) = a + b + bx + bx^2
\]
Where \( a \) and \( b \) are constants.
Checking Each Polynomial
- **(a) \( 5x^2 + 5x + 1 \)**
Can be written as \( 1 + 5x + 5x^2 \)
This is a linear combination: \( a = 1, b = 5 \).
- **(b) \( 100x^2 + 10x + 10 \)**
Can be expressed as \( 10 + 10x + 100x^2 \)
This does not satisfy the form \( a + b + bx + bx^2 \) since \( b \) cannot be 100.
- **(c) \( \pi(x^2 + 1) \)**
This can be written as \( \pi + \pi x^2 \)
It doesn't match the required form because there is no linear term.
- **(d) \( 7x^2 + \pi x + 3 \)**
This can be rewritten as \( 3 + \pi x + 7x^2 \)
This does not fit the form as \( b \) cannot be both \(\pi\) and 7 simultaneously.
Conclusion
The only polynomial that lies in the linear span of \( S \) is:
- **(a) \( 5x^2 + 5x + 1 \)**
To determine which polynomial lies in the linear span of \( S = \{1, 1 + x + x^2\} \), we need to check if each polynomial can be expressed as a linear combination of the elements in \( S \).
Elements of S
- The first element is \( 1 \).
- The second element is \( 1 + x + x^2 \).
Formulating the Linear Combination
A polynomial \( P(x) \) lies in the linear span of \( S \) if it can be expressed as:
\[
P(x) = a(1) + b(1 + x + x^2) = a + b + bx + bx^2
\]
Where \( a \) and \( b \) are constants.
Checking Each Polynomial
- **(a) \( 5x^2 + 5x + 1 \)**
Can be written as \( 1 + 5x + 5x^2 \)
This is a linear combination: \( a = 1, b = 5 \).
- **(b) \( 100x^2 + 10x + 10 \)**
Can be expressed as \( 10 + 10x + 100x^2 \)
This does not satisfy the form \( a + b + bx + bx^2 \) since \( b \) cannot be 100.
- **(c) \( \pi(x^2 + 1) \)**
This can be written as \( \pi + \pi x^2 \)
It doesn't match the required form because there is no linear term.
- **(d) \( 7x^2 + \pi x + 3 \)**
This can be rewritten as \( 3 + \pi x + 7x^2 \)
This does not fit the form as \( b \) cannot be both \(\pi\) and 7 simultaneously.
Conclusion
The only polynomial that lies in the linear span of \( S \) is:
- **(a) \( 5x^2 + 5x + 1 \)**
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Which of the following polynomials lies in the linear span of S = \{1, 1 + x + x ^ 2\} ? (a) 5x ^ 2 + 5x + 1 (b) 100x ^ 2 + 10x + 10 (c) pi(x ^ 2 + 1) (d) 7x ^ 2 + pi*x + 3?
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Which of the following polynomials lies in the linear span of S = \{1, 1 + x + x ^ 2\} ? (a) 5x ^ 2 + 5x + 1 (b) 100x ^ 2 + 10x + 10 (c) pi(x ^ 2 + 1) (d) 7x ^ 2 + pi*x + 3? for UPSC 2025 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Which of the following polynomials lies in the linear span of S = \{1, 1 + x + x ^ 2\} ? (a) 5x ^ 2 + 5x + 1 (b) 100x ^ 2 + 10x + 10 (c) pi(x ^ 2 + 1) (d) 7x ^ 2 + pi*x + 3? covers all topics & solutions for UPSC 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which of the following polynomials lies in the linear span of S = \{1, 1 + x + x ^ 2\} ? (a) 5x ^ 2 + 5x + 1 (b) 100x ^ 2 + 10x + 10 (c) pi(x ^ 2 + 1) (d) 7x ^ 2 + pi*x + 3?.
Which of the following polynomials lies in the linear span of S = \{1, 1 + x + x ^ 2\} ? (a) 5x ^ 2 + 5x + 1 (b) 100x ^ 2 + 10x + 10 (c) pi(x ^ 2 + 1) (d) 7x ^ 2 + pi*x + 3? for UPSC 2025 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Which of the following polynomials lies in the linear span of S = \{1, 1 + x + x ^ 2\} ? (a) 5x ^ 2 + 5x + 1 (b) 100x ^ 2 + 10x + 10 (c) pi(x ^ 2 + 1) (d) 7x ^ 2 + pi*x + 3? covers all topics & solutions for UPSC 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which of the following polynomials lies in the linear span of S = \{1, 1 + x + x ^ 2\} ? (a) 5x ^ 2 + 5x + 1 (b) 100x ^ 2 + 10x + 10 (c) pi(x ^ 2 + 1) (d) 7x ^ 2 + pi*x + 3?.
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