Advertisements
Advertisements
प्रश्न
If one of the zeroes of a quadratic polynomial of the form x2 + ax + b is the negative of the other, then it ______.
विकल्प
Has no linear term and the constant term is negative
Has no linear term and the constant term is positive
Can have a linear term but the constant term is negative
Can have a linear term but the constant term is positive
Advertisements
उत्तर
If one of the zeroes of a quadratic polynomial of the form x2 + ax + b is the negative of the other, then it has no linear term and the constant term is negative.
Explanation:
Let p(x) = x2 + ax + b
Put a = 0, then,
p(x) = x2 + b = 0
⇒ x2 = – b
⇒ `x = +- sqrt(-b)` ......[∴ b < 0]
Hence if one of the zeroes of quadratic polynomial p(x) is the negative of the other
Then it has no linear term
i.e., a = 0 and the constant term is negative
i.e., b < 0
Alternate Method:
Let f(x) = x2 + ax + b
And by given condition the zeroes area and – α
Sum of the zeroes = α – α = a
⇒ a = 0
f(x) = x2 + b, which cannot be linear,
and product of zeroes = α . (– α) = b
⇒ – α2 = b
which is possible when, b < 0
Hence, it has no linear term and the constant term is negative.
APPEARS IN
संबंधित प्रश्न
Find the zeros of the quadratic polynomial 4x2 - 9 and verify the relation between the zeros and its coffiecents.
Find a quadratic polynomial with the given numbers as the sum and product of its zeroes respectively.
`0, sqrt5`
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients
`q(x)=sqrt3x^2+10x+7sqrt3`
If 𝛼 and 𝛽 are the zeros of the quadratic polynomial f(x) = x2 − 5x + 4, find the value of `1/alpha+1/beta-2alphabeta`
If α and β are the zeros of the quadratic polynomial p(s) = 3s2 − 6s + 4, find the value of `alpha/beta+beta/alpha+2[1/alpha+1/beta]+3alphabeta`
If the zeros of the polynomial f(x) = x3 − 12x2 + 39x + k are in A.P., find the value of k.
Find the zeroes of the quadratic polynomial f(x) = 4x2 - 4x - 3 and verify the relation between its zeroes and coefficients.
Find the quadratic polynomial whose zeroes are `2/3` and `-1/4`. Verify the relation between the coefficients and the zeroes of the polynomial.
If 𝛼, 𝛽 are the zeroes of the polynomial f(x) = x2 + x – 2, then `(∝/β-∝/β)`
What should be added to the polynomial x2 − 5x + 4, so that 3 is the zero of the resulting polynomial?
If \[\sqrt{5}\ \text{and} - \sqrt{5}\] are two zeroes of the polynomial x3 + 3x2 − 5x − 15, then its third zero is
The polynomial which when divided by −x2 + x − 1 gives a quotient x − 2 and remainder 3, is
If the zeroes of a quadratic polynomial ax2 + bx + c are both positive, then a, b and c all have the same sign.
If two of the zeroes of a cubic polynomial are zero, then it does not have linear and constant terms.
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:
`2s^2 - (1 + 2sqrt(2))s + sqrt(2)`
For the following, find a quadratic polynomial whose sum and product respectively of the zeroes are as given. Also find the zeroes of these polynomials by factorisation.
`-2sqrt(3), -9`
Given that the zeroes of the cubic polynomial x3 – 6x2 + 3x + 10 are of the form a, a + b, a + 2b for some real numbers a and b, find the values of a and b as well as the zeroes of the given polynomial.
Given that `sqrt(2)` is a zero of the cubic polynomial `6x^3 + sqrt(2)x^2 - 10x - 4sqrt(2)`, find its other two zeroes.
If α, β are zeroes of the quadratic polynomial x2 – 5x + 6, form another quadratic polynomial whose zeroes are `1/α, 1/β`.
The zeroes of the polynomial p(x) = 25x2 – 49 are ______.
