Question

Find quadriatic polynomial whose zeros are -4 and -5

Answer 1

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

Zeroes of required polynomial are given as -4 and -5.

$\large\bf\underline {To \: find:-}$

• The quadratic polynomial.

$\huge\bf\underline{Solution:-}$

Let α and β are the zeroes of the required polynomial.

• Let α be -4 and β be -5.

Sum of zeroes :-

$\rm \rightarrowtail \alpha + \beta = - 4 + ( - 5) \\ \\ \rightarrowtail \rm \: \alpha + \beta = - 4 - 5 \\ \\ \rightarrowtail \rm \: \alpha + \beta = - 9$

Product of zeroes :-

$\rightarrowtail \rm \: \alpha \beta = - 4 \times ( - 5) \\ \\ \rightarrowtail \rm \: \alpha \beta = 20$

Formula for quadratic polynomial:-

$\bigstar \bf \: {x}^{2} - ( \alpha + \beta )x + \alpha \beta$

$\dashrightarrow \rm \: {x}^{2} - ( - 9)x + (20) \\ \\ \dashrightarrow \rm \: {x}^{2} + 9x + 20 \\ \\ \bullet \bf \:quadratic \: polynomial \:is = {x}^{2} + 9x + 20$

$\large\underline{\bf\dag \: Verification:-}$

• p(x) = x² + 9x + 20

• a = 1
• b = 9
• c = 20

$\ddag \rm \: sum \: of \: zeroes = \frac{ - b}{a} \\ \\ \longrightarrow \rm \: - 9 = \frac{ - 9}{1} \\ \\ \longrightarrow \rm \: - 9 = - 9 \\ \\ \ddag \rm \: product\: of \: zeroes = \frac{ c}{a} \\ \\ \longrightarrow \rm \:20 = \frac{20}{1} \\ \\ \longrightarrow \rm \:20 = 20$

LHS = RHS

hence Verified

Answer 2

✯✯ QUESTION ✯✯

Find quadriatic polynomial whose zeros are -4 and -5 ..

━━━━━━━━━━━━━━━━━━━━

✰✰ ANSWER ✰✰

$\implies\tt{Let\:\alpha\:of\:Zeroes=-4}$

$\implies\tt{Let\:\beta\:of\:Zeroes=-5}$

➥Now ,

★Sum Of Zeroes : -

$\implies\tt{\alpha+\beta=-4+(-5)}$

$\implies\tt{\alpha+\beta=-9}$

★Product Of Zeroes : -

$\implies\tt{\alpha\beta=-4\times{(-5)}}$

$\implies\tt{\alpha\beta=20}$

➥Using Formula : -

$\implies\tt{\large{\boxed{\bold{\bold{\orange{\sf{{x}^{2}-(\alpha+\beta)x+(\alpha\beta)}}}}}}}$

➥Putting Values : -

$\implies\tt{{x}^{2}-(-9)x+(20)}$

$\implies\tt{{x}^{2}+9x+20}$

➮So , the quadratic polynomial is x² + 9x + 20 ..

_______________________

➮VERIFICATION : -

$\implies\tt{{x}^{2}+9x+20}$

➥Here : -

• a ⇒ 1
• b ⇒ 9
• c ⇒ 20

★Sum Of Zeroes : -

$\implies\tt{\dfrac{-b}{a}=\dfrac{-(9)}{1}}$

$\implies\tt{-9=-9}$

$\red\longmapsto\:\large\underline{\boxed{\bf\green{L.H.S}\orange{=}\purple{R.H.S}}}$

★Product Of Zeroes : -

$\implies\tt{\dfrac{c}{a}=\dfrac{20}{1}}$

$\implies\tt{20=20}$

$\purple\longmapsto\:\large\underline{\boxed{\bf\orange{L.H.S}\green{=}\red{R.H.S}}}$

✺ HENCE VERIFIED ✺