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If 2 and-2 are the zeroes of the polynomial p(x) =ax^4+2x^3-3x^2+bx-4 find the values of a and n and hence p(-3)?
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If 2 and-2 are the zeroes of the polynomial p(x) =ax^4+2x^3-3x^2+bx-4 ...
Given Information:
- Zeroes of the polynomial p(x) are 2 and -2
- p(x) = ax^4 + 2x^3 - 3x^2 + bx - 4
Finding the values of a and b:
- Since 2 and -2 are zeroes of p(x), the factors of p(x) are (x-2) and (x+2)
- Therefore, p(x) = a(x-2)(x+2)(x-r)(x-s) where r and s are the other two zeroes
- Expanding p(x) using the given polynomial, we get: p(x) = ax^4 + 2x^3 - 3x^2 + bx - 4
- Equating the coefficients of corresponding terms in both expressions, we get:
1. a = 1 (coefficient of x^4 term)
2. -4a + 2 = -3 (coefficient of x^2 term)
3. 4a - 4b = 0 (coefficient of x term)
- Solving the above equations, we find a = 1 and b = 1
Finding the other two zeroes:
- Using the sum and product of zeroes formula for a quadratic equation, we get:
1. Sum of zeroes = -b/a = -2
2. Product of zeroes = c/a = 4
- Solving the above equations, we find the other two zeroes to be 1 and -4
Calculating p(-3):
- Substituting x = -3 in p(x), we get:
p(-3) = 1(-3)^4 + 2(-3)^3 - 3(-3)^2 + 1(-3) - 4
p(-3) = 81 - 54 - 27 - 3 - 4
p(-3) = -7
Therefore, the values of a and b are 1 and 1 respectively, and p(-3) = -7.
- Zeroes of the polynomial p(x) are 2 and -2
- p(x) = ax^4 + 2x^3 - 3x^2 + bx - 4
Finding the values of a and b:
- Since 2 and -2 are zeroes of p(x), the factors of p(x) are (x-2) and (x+2)
- Therefore, p(x) = a(x-2)(x+2)(x-r)(x-s) where r and s are the other two zeroes
- Expanding p(x) using the given polynomial, we get: p(x) = ax^4 + 2x^3 - 3x^2 + bx - 4
- Equating the coefficients of corresponding terms in both expressions, we get:
1. a = 1 (coefficient of x^4 term)
2. -4a + 2 = -3 (coefficient of x^2 term)
3. 4a - 4b = 0 (coefficient of x term)
- Solving the above equations, we find a = 1 and b = 1
Finding the other two zeroes:
- Using the sum and product of zeroes formula for a quadratic equation, we get:
1. Sum of zeroes = -b/a = -2
2. Product of zeroes = c/a = 4
- Solving the above equations, we find the other two zeroes to be 1 and -4
Calculating p(-3):
- Substituting x = -3 in p(x), we get:
p(-3) = 1(-3)^4 + 2(-3)^3 - 3(-3)^2 + 1(-3) - 4
p(-3) = 81 - 54 - 27 - 3 - 4
p(-3) = -7
Therefore, the values of a and b are 1 and 1 respectively, and p(-3) = -7.
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If 2 and-2 are the zeroes of the polynomial p(x) =ax^4+2x^3-3x^2+bx-4 find the values of a and n and hence p(-3)?
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If 2 and-2 are the zeroes of the polynomial p(x) =ax^4+2x^3-3x^2+bx-4 find the values of a and n and hence p(-3)? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about If 2 and-2 are the zeroes of the polynomial p(x) =ax^4+2x^3-3x^2+bx-4 find the values of a and n and hence p(-3)? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If 2 and-2 are the zeroes of the polynomial p(x) =ax^4+2x^3-3x^2+bx-4 find the values of a and n and hence p(-3)?.
If 2 and-2 are the zeroes of the polynomial p(x) =ax^4+2x^3-3x^2+bx-4 find the values of a and n and hence p(-3)? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about If 2 and-2 are the zeroes of the polynomial p(x) =ax^4+2x^3-3x^2+bx-4 find the values of a and n and hence p(-3)? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If 2 and-2 are the zeroes of the polynomial p(x) =ax^4+2x^3-3x^2+bx-4 find the values of a and n and hence p(-3)?.
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