Question

Find a cubic polynomial with the sum, sum of product of its zeroes taken two at a time, and productof its zeroes as 2, - 7, - 14 respectively.

Answer 1

Step-by-step explanation:

i hope this easy formula trick will help u

Answer 2

$\large\bf\underline{Given:-}$

• sum of zeroes = 2
• Sum of product of zeroes taken two at a time = -7
• product of zeroes = -14

$\large\bf\underline {To \: find:-}$

• cubic polynomial

$\huge\bf\underline{Solution:-}$

Let α , β and γ be the zeroes of the required polynomial.

• α+β+γ = 2
• αβ + βγ + γα = -7
• αβγ = -14

Formula for cubic polynomial:-

$\small \bf \: {x}^{3} - ( \alpha + \beta + \gamma ) {x}^{2} + ( \alpha \beta + \beta \gamma + \gamma \alpha )x - \alpha \beta \gamma$

$\rm \mapsto \: {x}^{3} - (2) {x}^{2} + ( - 7)x - ( - 14) \\ \\ \bf \mapsto \: {x}^{3} - 2 {x}^{2} - 7x + 14$

So,

the required polynomial is x³-2x² -7x + 14

$\bf\underline{Verification:-}$

• p(x) = x³-2x² -7x + 14

where,

• a = 1
• b = -2
• c = -7
• d = 14

we know that,

• α+β+γ = - b/a

➝ 2 = -(-2)/1

➝ 2 = 2

• αβ + βγ + γα = c/a

➝ -7 = -7/1

➝ -7 = -7

• αβγ = -d/a

➝ -14 = -14/1

➝ -14 = -14

LHS = RHS

hence Verified