Question

if 2,5 are zeroes of polynomial p(x), then find p(x)​

Answer 1

$\large\bf\underline \blue {To \: \mathscr{f}ind:-}$

• we need to find the polynomial.

$\huge\bf\underline \red{ \mathcal{S}olution:-}$

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

• Zeroes of required polynomial are 2 and 5

Let the zeroes be α and β

• Let α = 2
• and β = 5

✫ Sum of zeroes = α + β

⇝α + β = 2 + 5

⇝α + β = 7

✫ Product of zeroes = αβ

⇝αβ = 2 × 5

⇝αβ = 10

• By using quadratic equation Formula :-

• x² -(α + β) x + αβ

⇝ x² -(7)x + 10

⇝ x² - 7x + 10

Hence,

• Required polynomial is x² - 7x + 10

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Answer 2

$\bf \huge \red{answer }:$

$\bf \green{ f(2) = 0} \: \: \: \: \: \: \: \: \: \: \: \: \green{f(5) = 0}$

$\bf \huge \fbox \red { \: factor \: therom \: }$

If p(a)=0 then reminder also zero and so (x-a) is a factor of p(x)

$\bf \: f(2) = 0 \: \: \: \: \: \: \: \: \: \: \: f(5) = 0$

$\bf \: (x - 2)(x - 5) = 0$

$\bf \: x(x - 5) - 2(x - 5) = 0$

$\bf \: {x}^{2} - 5x - 2x + 10 = 0$

$\bf \: {x}^{2} - 7x + 10 = 0$

$\bf \green { \therefore \: the \: required \: polynomial \: is \: {x}^{2} - 7x + 10 }$