Question

Find a cubic palynomial whose zeroes are -2 -3 and -1 ​

Answer 1

$\large\bf\underline \blue {To \: \mathscr{f}ind:-}$

• we need to find the cubic polynomial.

$\huge\bf\underline \purple{ \mathcal{S}olution:-}$

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

• zero of required cubic polynomial are -2 , -3 and -1

${ \blue{ \mathscr{ \underline{Let : - }}}}$

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

• Let α = -2
• β = -3
• Ɣ = -1

⇝(α + β + Ɣ) = -2 + (-3) + (-1)

⇝(α + β + Ɣ) = -2 - 3 - 1

⇝(α + β + Ɣ) = -6 ⠀....2)

⇝(αβ + βƔ + Ɣα) = -2 × (-3) + (-3) × (-1) + (-1) ×(-2)

⇝(αβ + βƔ + Ɣα) = 6 + 3 + 2

⇝(αβ + βƔ + Ɣα) = 11 ⠀....1)

⇝ (αβƔ) = -2 × (-3) × (-1)

⇝ (αβƔ) = -6 ⠀....3)

• Formula for quadratic polynomial is:-

▶x³ -(α + β + Ɣ)x² + (αβ + βƔ + Ɣα)x -(αβƔ)

Putting values of (α + β + Ɣ), (αβ + βƔ + Ɣα) and (αβƔ) in the formula .

↛x³ -(-6)x² + (11)x - (-6)

↛x³ + 6x² + 11x + 6

Hence

The required cubic polynomial is:-

• x³ + 6x² + 11x + 6

━━━━━━━━━━━━━━━━━━━━━━━━━

Answer 2

$\bf\huge\underline\red{answer} : -$

$\bf\underline \blue{standard \: form \: for \: cubic \: polynomial} :$

$\implies \bf \orange{ {ax}^{3} + {bx}^{2} + cx + d}$

OR

$\implies \bf \ {x}^{3} - ( \alpha + \beta + \gamma ) {x}^{2} + ( \alpha \beta + \beta \gamma + \gamma \alpha )x - ( \alpha \beta \gamma )........(1)$

$\bf \large \ \: its \: zeroes \: are \: \alpha , \: \beta \: and \: \gamma$

$\bf \: \: {\alpha = - 2}$

$\bf \: \beta = - 3$

$\bf \: \gamma = - 1$

Now,

$\bf \green{ \alpha + \beta + \gamma = - 2 - 3 - 1 = - 6}$

$\bf \green{ \alpha \beta \gamma = ( - 2)( - 3)( - 1) = - 6}$

$\bf \green{ \alpha \beta + \beta \gamma + \gamma \alpha = ( - 2)( - 3) + ( - 3)( - 1) + ( - 1)( - 2) = 6 + 3 + 2 = 11}$

substitute these values in eq.(1)

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

$\implies \bf \: {x}^{3} - ( - 6) {x}^{2} + (11)x - ( - 6)$

$\implies \bf \: {x}^{3} + 6 {x}^{2} + 11x + 6$

$\bf \huge \red{cubic \: polynomial \: is \: {x}^{3} + 6 {x}^{2} + 11x + 6}$