Question

If one of the zeroes of a quadratic polynomial of the form x²+ax+ b is the negative of the other, then it :
(a) has no linear term and the constant ferm is negative
(b) has no linear term and the constant term is positive
(c) can have a linear term but the constant term is negative.
(d) can have a linear term but the constant term is positive​

Answer 1

Answer:

please click on brainliest

Step-by-step explanation:

Let p(x) = x2 + ax + b.

Put a = 0, then, p(x) = x2 + b = 0

⇒ x2 = -b

⇒ x = ± ±√-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 = O 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.

Answer 2

SOLUTION

TO CHOOSE THE CORRECT OPTION

If one of the zeroes of a quadratic polynomial of the form x² + ax + b is the negative of the other, then it

(a) has no linear term and the constant ferm is negative

(b) has no linear term and the constant term is positive

(c) can have a linear term but the constant term is negative.

(d) can have a linear term but the constant term is positive

EVALUATION

Here the given polynomial is x² + ax + b

Since one of the zeroes is the negative of the other

Let the zeroes are β and - β

Then we have

$\displaystyle \sf{ Sum \: of \: the \: zeroes = - \frac{a}{1} }$

$\displaystyle \sf{ \implies \beta - \beta = - \frac{a}{1} }$

$\boxed{ \: \: \displaystyle \sf{ \implies a = 0 } \: \: }$

$\displaystyle \sf{ Product \: of \: the \: zeroes = \frac{b}{1} }$

$\displaystyle \sf{ \implies \: \beta \times ( - \beta ) = b }$

$\displaystyle \sf{ \implies \: b = - { \beta }^{2} }$

$\boxed{ \: \: \displaystyle \sf{ \implies \: b < 0 } \: \: }$

So the polynomial has no linear term and the constant ferm is negative

FINAL ANSWER

Hence the correct option is

(a) has no linear term and the constant ferm is negative

━━━━━━━━━━━━━━━━

Learn more from Brainly :-

$1.$ find the equation that formed by increasing each root of 2x²-3x-1=0by 1

https://brainly.in/question/33063519

2. find the equation that formed by squaring each root of the equation x²+3x-2=0

https://brainly.in/question/33064705