Question

Two zeroes of cubic polynomial ax³ + 3x² – bx – 6 are –1 and –2. Find the third zero and value of a & b.

Answer 1

${ \bold{ \boxed{ \boxed{ \red{ \mathtt{ \bigstar \: ANSWER \: \bigstar}}}}}}$

${ \bold{ \underline{Given \: \: polynomial} : - }} \\ \\ { \bold{ \orange{ \mathtt{f(x) = a {x}^{3} + 3 {x}^{2} - bx - 6}}}} \\ \\ { \bold{ \orange{ \mathtt{it's \: \: two \: \: roots \: \: are \: \: - 1 \: \: and \: \: - 2}}}} \\ \\ {\bold{ \orange{ \mathtt{so \: \: that - }}}} \\ \\ {\bold{ \orange{ \mathtt{f( - 1) = - a + 3 + b - 6 = 0 \: \: \: ...(1)}}}} \\ \\ {\bold{ \orange{ \mathtt{f( - 2) = - 8a + 12 + 2b - 6 = 0 \: ...(2)}}}} \\ \\ {\bold{ \orange{ \mathtt{by \: \: (1) \: \: and \: \: (2) - }}}} \\ \\ {\bold{ \orange{ \mathtt{a = 2 \: \: and \: \: b = 5}}}} \\ \\ {\bold{ \pink{ \mathtt{so \: that - }}}} \\ \\ {\bold{ \orange{ \mathtt{f(x) = 2 {x}^{3} + 3 {x}^{2} - 5x - 6 }}}} \\ \\ {\bold{ \red{ \mathtt{we \: should \: \: write \: \: this \: \: equation \: \: as - }}}} \\ \\ {\bold{ \orange{ \mathtt{f(x) = 2(x + 1)(x + 2)(x - \alpha )}}}} \\ \\ {\bold{ \blue{ \mathtt{here \: \: \alpha \: \: is \: \: another \: \: root - }}}} \\ \\ {\bold{ \red{ \mathtt{let's \: \: multiply \: \: and \: \: compare - }}}} \\ \\ {\bold{ \orange{ \mathtt{ : \implies \: f(x) = 2 {x}^{3} - 2 \alpha {x}^{2} + 4 {x}^{2} - 4 \alpha x + 2 { {x}^{2} }^{} - 2 \alpha x + 4x - 4 \alpha }}}} \\ \\ {\bold{ \huge{ \orange :{\implies \boxed{ \orange{ \mathtt{ \alpha = \frac{3}{2} }}}}}}}$