Question

If all the zeroes of a cubic polynomial are negative, then the sign of coefficient of each term and the constant term have the same sign. state true or false. give reason for your answer.​

Answer 1

Answer:

the bag of marbles was left from unknown

Answer 2

Step-by-step explanation:

Given: If all the zeroes of a cubic polynomial are negative, then the sign of coefficient of each term and the constant term have the same sign.

To find: state true or false. give reason for your answer.

Solution:

True

Justification:

Let $\alpha,\beta$ and $\gamma$

are the zeros of cubic polynomial,then relationship between zeros and coefficient of x³,x²,x and constant term is given by

$\alpha + \beta + \gamma = \frac{ - b}{a}...eq1 \\ \\ \alpha \beta + \beta \gamma + \alpha \gamma = \frac{c}{a}...eq2 \\ \\ \alpha \beta \gamma = \frac{ - d}{a}...eq3$

ATQ, all the zeroes of a cubic polynomial are negative.

Let say,

$\bold{\green{- \alpha}} \\ \bold{\orange{ - \beta }} \\\bold{\pink{ -\gamma}}$

put these values of $\alpha,\beta$ in eq1,eq2 and eq3

$-\alpha + (- \beta) + (- \gamma) = \frac{ - b}{a}\\ \\-(\alpha + \beta + \gamma) = \frac{ - b}{a} \\ \\\bold{\red{\alpha + \beta + \gamma=\frac{b}{a}}}$

$\alpha \beta + \beta \gamma + \alpha \gamma = \frac{c}{a} \\ \\(- \alpha)(-\beta) + (-\beta)( -\gamma) + (-\alpha) (- \gamma) = \frac{c}{a} \\ \\ \bold{\red{\alpha \beta + \beta \gamma + \alpha \gamma = \frac{c}{a} }}$

and

$\alpha \beta \gamma = \frac{ - d}{a} \\ \\ (-\alpha) (-\beta) (- \gamma) = \frac{ - d}{a} \\ \\ - \alpha \beta \gamma = \frac{ - d}{a}\\ \\ \bold{\red{ \alpha \beta \gamma = \frac{d}{a}}}$

Therefore it is true that, If all the zeroes of a cubic polynomial are negative, then the sign of coefficient of each term and the constant term have the same sign.

Final answer:

True

Hope it helps you.

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