Question

check 3 and -2 are zeroes of p(x) x2-x-6 and verify the relationship between the zeroes and the coefficients​

Answer 1

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀☆⠀GIVEN  POLYNOMIAL :  x² - x - 6 = 0

⠀⠀⠀⠀⠀Checking whether 3 and -2 are zeroes of polynomial or not .

$\qquad \dashrightarrow \sf x^2 - x - 6 \: =\:\:0\: \: \\\\\qquad \dashrightarrow \sf x^2 - 3x + 2x - 6 \: =\:\:0\: \: \\\\\qquad \dashrightarrow \sf x (x - 3) + 2 ( x - 3 ) \: =\:\:0\: \: \\\\\qquad \dashrightarrow \sf (x - 3) ( x + 2 ) \: =\:\:0\: \: \\\\\qquad \dashrightarrow \sf \:x\:=\:\: 3 \:\:or\:\:x\:\:=\:\: - 2 \:\:\: \: \\\\\qquad \therefore \underline {\boxed{\pmb{\purple{\frak{\:x\:= \: 3 \:\: or \: -2 \:\:\: }}}}}\:\:\bigstar$

$\qquad \therefore \:\underline { \sf The \:zeroes \:of \:Polynomial \:are \: \bf 3 \:\: \sf and \:\bf -2 \:\:.}$

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀

$\qquad \bigstar \:\:\underline {\sf \: Verifying \: \:\bf \:Relationship\: \:\sf\:between \:\:\bf\:zeroes\:\sf\:and\:\bf Cofficients \:\sf\:\:of \:\:\:Polynomial \:\:}:$

$\qquad \underline {\boxed {\pmb {\red { \:\maltese \:\: Sum \: of \: zeroes \:}\: \:\purple{ ( \: \alpha \:+\beta \:)}\red{\:\::\:}}}}\\\\\qquad \dashrightarrow \sf \bigg( \:\: \alpha \:\:+ \: \beta \:\:\bigg) \:=\: \dfrac{-(Cofficient \:of\:x)\:}{Cofficient \:of \:x^2 \:\:}\:\:\\\\\qquad \dashrightarrow \sf \bigg( \:\: 3\:\:\bigg) \:+\:\bigg( \:\:-2 \:\:\bigg) \:=\: \dfrac{-(-1)\:}{1 \:\:}\:\:\\\\\qquad \dashrightarrow \sf \bigg( \:\: 3\:\:- \:\:2 \:\:\bigg) \:=\: \dfrac{-(-1)\:}{1 \:\:}\:\:\\\\\qquad \dashrightarrow \sf \bigg( \:\: 3\:\:- \:\:2 \:\:\bigg) \:=\: \dfrac{1\:}{1 \:\:}\:\:\\\\\qquad \dashrightarrow \sf \bigg( \:\: 1 \:\:\bigg) \:=\: \dfrac{1\:}{1 \:\:}\:\:\\\\\qquad \dashrightarrow \sf \:\: 1 \: \:=\: 1\:\:\\\\\qquad \therefore \underline {\boxed{\pmb{\purple{\frak{\:1 \:\:= \:\:1\: }}}}}\:\:\bigstar$

⠀⠀⠀⠀⠀AND ,

$\qquad \underline {\boxed {\pmb {\red { \:\maltese \:\: Product \: of \: zeroes \:}\: \:\purple{ ( \: \alpha \:\beta \:)}\red{\:\::\:}}}}\\\\\qquad \dashrightarrow \sf \bigg( \:\: \alpha \:\: \: \beta \:\:\bigg) \:=\: \dfrac{ \:Constant \:Term\:}{Cofficient \:of \:x^2 \:\:}\:\:\\\\\qquad \dashrightarrow \sf \bigg( \:\: 3\:\:\bigg) \:\times\:\bigg( \:\:-2 \:\:\bigg) \:=\: \dfrac{-6\:}{1 \:\:}\:\:\\\\\qquad \dashrightarrow \sf \:\: 3\:\: \:\times\:\bigg( \:\:-2 \:\:\bigg) \:=\: \dfrac{-6\:}{1 \:\:}\:\:\\\\\qquad \dashrightarrow \sf \:\: 3\:\: \:\times\:\bigg( \:\:-2 \:\:\bigg) \:=\: -6\:\:\:\\\\\qquad \dashrightarrow \sf \:\:\:\bigg( \:\:-6 \:\:\bigg) \:=\: -6\:\:\:\\\\\qquad \dashrightarrow \sf \:\:\: \:\:-6 \:\: \:=\: -6\:\:\:\\\\\qquad \therefore \underline {\boxed{\pmb{\purple{\frak{\:-6 \:\:= \:\:-6\: }}}}}\:\:\bigstar$

$\qquad \therefore \:\:\underline {\pmb{\bf { Hence \:,\:Verified \:!\;}}}$