Question

if the product of two zeros of the polynomial is 2 x cube + 6 x square - 4 x + 9 is 3 then find its third zero

Answer 1

$\underline{\mathfrak{ Solution : }}$

$\textsf{ A cubic polynomial has generally } \\ \mathsf{ its \: zeroes \: as \: \alpha , \beta \: and \: \gamma. }$

$\textsf{ We know relationship between } \\ \textsf{coefficient of a cubic polynomial } \\ \textsf{and its zeroes. }$

$\mathsf{ \implies Sum \: of \: zeroes \: = \: \dfrac{-b}{a}}\\ \\ \mathsf{\implies Sum \: of \: zeroes \: taken \: two \: at \: a \: time \: = \: \dfrac{c}{a} \: } \\ \\ \mathsf{ \implies Product \: of \: zeroes \: = \: \dfrac{-d}{a}}$

$\\ \mathsf{ Where, } \\ \\ \mathsf{ \longrightarrow a \: = \: Coefficient \: of \: {x}^{3}} \\ \\ \mathsf{\longrightarrow b \: = \: Coefficient \: of \: {x}^{2}}\\ \\ \mathsf{\longrightarrow c \: = \: Coefficient \: of \: x } \\ \\ \mathsf{ \longrightarrow d \: = \: Constant \: term }$

$\mathsf{ Given \: Polynomial : } \\ \\ \mathsf{ = 2{x}^{3} \: + \: 6{x}^{2} \: - \: 4x \: + \: 9 }$

$\mathsf{Here, } \\ \\ \mathsf{ \longrightarrow a \: = \: 2 } \\ \\ \mathsf{\longrightarrow b \: = \: 6 } \\ \\ \mathsf{\longrightarrow c \: = \: -4 } \\ \\ \mathsf{\longrightarrow d \: = \: 9 }$

$\mathsf{ Now,} \\ \\ \mathsf{ In \: question \: product \: of \: two \: zeroes } \\ \mathsf{are \: already \: given \: , \: so \: we \: can \: easily} \\ \mathsf{find \: the \: third \: zero \: by \: relation \: of \: } \\ \mathsf{product \: of \: zeroes \: and \: coefficients} \\ \mathsf{o f \: polynomial.}$

$\mathsf{ Now, } \\ \\ \mathsf{ \implies Product \: of \: zeroes \: = \: \dfrac{-d}{a}} \\ \\ \mathsf{ Let \: third \: zero \: is \: \gamma. } \\ \\ \mathsf{ \implies 3 \gamma\: = \: \dfrac{ - 9}{2} } \\ \\ \mathsf{ \implies \gamma \: = \: \dfrac{ - 9}{ \: 2 \: \times \: 3 \: } } \\ \\ \mathsf{ \therefore \: \: \gamma \: = \: \dfrac{ - 3}{2} }$

$\boxed{\underline{ \: \mathfrak{ Hope \: \: it \: \: helps \: \: !! } \: }}$

Answer 2

here is your answer....