Question

factorize the cubic polynomial 5xcube+2xsquare - 13x+6​

Answer 1

$\large \boxed{{\boxed{ \underline{ \underline \blue{ \sf{answer : }}}}}}$

$\sf{5x {}^{3} + 2x {}^{2} - 13x + 6 }$

$\sf{by \: hit \: and \: trial \: method : } \\ \sf{ put\: x = 1 \: in \: the \: given \: polynomial} \\ \sf{p(x) = 5x {}^{3} + 2x {}^{2} - 13x + 6 } \\ \sf{p(1) = 5(1) {}^{3} + 2(1) {}^{2} - 13(1) + 6 } \\ \sf{p(1) = 5 + 2 - 13 + 6} \\ \sf{p(1) = 13 - 13} \\ \sf{p(1) = 0}$

$\sf{x = 1 \: is \: a \: zero \: of \: p(x)} \\ \sf{(x - 1) \: is \:factor \: of \: p(x) }$

$\sf{dividing \: p(x) \: by \: (x - 1)}$

$\sf{ x - 1 \: \: \overline{ ) \: \cancel{5x {}^{3} }+ 2x {}^{2} - 13x + 6 \: ( } \: \: \: 5x {}^{2} + 7x - 6 } \\ \sf{ \underline{ \cancel{- 5x {}^{3} }+ 5x } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: } \\ \sf{ \cancel{7x {}^{2} }- 13x \: \: \: \: \: \: \: \: \: \: \: \: \: } \\ \sf{ \underline{ \cancel{ - 7x {}^{2} }+ 7x} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: } \\ \cancel{\sf{ - 6x + 6}} \\ \underline{ \cancel{\sf{ - 6x + 6}}} \\ \underline{\sf{ \: \: \: \: \: \: 0 \: \: \: \: \: \: \: }}$

$\sf{q(x) = 5x {}^{2} + 7x - 6 }$

$\sf{by \: factorising \: q(x) : }$

$\sf{5x {}^{2} + 7x - 6 } \\ \sf{5x {}^{2} + 10x - 3x - 6 } \\ \sf{5x(x + 2) - 3(x + 2)} \\ \sf{(x + 2)(5x - 3)}$

$\sf{hence} \\ \sf{(x - 1)(x + 2)(5x - 3) } \\ \sf{are \: the \: factors \: of \: the \: given \: polynomial}$