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If one of the zeros of the cubic polynomial x³ ax² bx c is -1 then the product of the other two zeros is ?? a) a - b - 1 b) b - a - 1 c) 1 - a b d) 1 a - b Explain also.?
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If one of the zeros of the cubic polynomial x³ ax² bx c is -1 th...
(a) Let p(x) = x3 + ax2 + bx + c
Let a, p and y be the zeroes of the given cubic polynomial p(x).
∴ α = -1 [given]
and p(−1) = 0
⇒ (-1)3 + a(-1)2 + b(-1) + c = 0
⇒ -1 + a- b + c = 0
⇒ c = 1 -a + b …(i)
We know that,
αβγ = -c
⇒ (-1)βγ = −c [∴α = -1]
⇒ βγ = c
⇒ βγ = 1 -a + b [from Eq. (i)]
Hence, product of the other two roots is 1 -a + b.
Alternate Method
Since, -1 is one of the zeroes of the cubic polynomial f(x) = x2 + ax2 + bx + c i.e., (x + 1) is a factor of f{x).
Now, using division algorithm,
⇒x3 + ax2 + bx +c = (x + 1) x {x2 + (a – 1)x + (b – a + 1)> + (c – b + a -1)
⇒x3 + ax2 + bx + (b – a + 1) = (x + 1) {x2 + (a – 1)x + (b -a+ 1)}
Let a and p be the other two zeroes of the given polynomial, then
Let a, p and y be the zeroes of the given cubic polynomial p(x).
∴ α = -1 [given]
and p(−1) = 0
⇒ (-1)3 + a(-1)2 + b(-1) + c = 0
⇒ -1 + a- b + c = 0
⇒ c = 1 -a + b …(i)
We know that,
αβγ = -c
⇒ (-1)βγ = −c [∴α = -1]
⇒ βγ = c
⇒ βγ = 1 -a + b [from Eq. (i)]
Hence, product of the other two roots is 1 -a + b.
Alternate Method
Since, -1 is one of the zeroes of the cubic polynomial f(x) = x2 + ax2 + bx + c i.e., (x + 1) is a factor of f{x).
Now, using division algorithm,
⇒x3 + ax2 + bx +c = (x + 1) x {x2 + (a – 1)x + (b – a + 1)> + (c – b + a -1)
⇒x3 + ax2 + bx + (b – a + 1) = (x + 1) {x2 + (a – 1)x + (b -a+ 1)}
Let a and p be the other two zeroes of the given polynomial, then
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If one of the zeros of the cubic polynomial x³ ax² bx c is -1 then the product of the other two zeros is ?? a) a - b - 1 b) b - a - 1 c) 1 - a b d) 1 a - b Explain also.?
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If one of the zeros of the cubic polynomial x³ ax² bx c is -1 then the product of the other two zeros is ?? a) a - b - 1 b) b - a - 1 c) 1 - a b d) 1 a - b Explain also.? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about If one of the zeros of the cubic polynomial x³ ax² bx c is -1 then the product of the other two zeros is ?? a) a - b - 1 b) b - a - 1 c) 1 - a b d) 1 a - b Explain also.? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If one of the zeros of the cubic polynomial x³ ax² bx c is -1 then the product of the other two zeros is ?? a) a - b - 1 b) b - a - 1 c) 1 - a b d) 1 a - b Explain also.?.
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