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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
Community Answer
If one of the zeros of the cubic polynomial x³ ax² bx c is -1 th...
Understanding the Cubic Polynomial
A cubic polynomial can be represented as:
P(x) = x³ + ax² + bx + c
Given that one of the zeros, α, is -1, we can use this information to analyze the polynomial further.
Relationship Between Zeros
Let the zeros of the polynomial be α, β, and γ. According to Vieta's formulas, we know the following relationships hold:
- The sum of the zeros: α + β + γ = -a
- The sum of the products of the zeros taken two at a time: αβ + αγ + βγ = b
- The product of the zeros: αβγ = -c
Substituting the Known Zero
Since α = -1, we can substitute this into the relationships:
- -1 + β + γ = -a
This simplifies to: β + γ = -a + 1
Finding the Product of the Other Zeros
To determine the product βγ, we can use the product of the zeros formula:
- (-1) * β * γ = -c
Thus, βγ = c - 1
Now, we also have the equation from the sum of the products:
- (-1)β + (-1)γ + βγ = b
This leads to: -β - γ + βγ = b
Substituting β + γ = -a + 1 into this equation gives:
-[-a + 1] + βγ = b
This implies:
βγ = b + a - 1
Conclusion
Thus, the product of the other two zeros (β and γ) is:
b + a - 1
Therefore, the correct option is (a - b - 1), which occurs when rearranged from the derived expression.
A cubic polynomial can be represented as:
P(x) = x³ + ax² + bx + c
Given that one of the zeros, α, is -1, we can use this information to analyze the polynomial further.
Relationship Between Zeros
Let the zeros of the polynomial be α, β, and γ. According to Vieta's formulas, we know the following relationships hold:
- The sum of the zeros: α + β + γ = -a
- The sum of the products of the zeros taken two at a time: αβ + αγ + βγ = b
- The product of the zeros: αβγ = -c
Substituting the Known Zero
Since α = -1, we can substitute this into the relationships:
- -1 + β + γ = -a
This simplifies to: β + γ = -a + 1
Finding the Product of the Other Zeros
To determine the product βγ, we can use the product of the zeros formula:
- (-1) * β * γ = -c
Thus, βγ = c - 1
Now, we also have the equation from the sum of the products:
- (-1)β + (-1)γ + βγ = b
This leads to: -β - γ + βγ = b
Substituting β + γ = -a + 1 into this equation gives:
-[-a + 1] + βγ = b
This implies:
βγ = b + a - 1
Conclusion
Thus, the product of the other two zeros (β and γ) is:
b + a - 1
Therefore, the correct option is (a - b - 1), which occurs when rearranged from the derived expression.
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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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