Question

$if \: x - 4 \: \: is \: a \: factor \: of \: {x}^{3} - {5x}^{2} - px + 24 \: find \: the \: value \: of \: p$
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Answer 1

ATQ, x - 4 is a factor of polynomial x³ - 5x² - px + 24

i.e remainder is 0 since when polynomials are divided by their factors, remainder comes 0.

given :-

x - 4 = 0

➡ x = 4

putting value of x in the polynomial.

➡ p(4) = (4)³ - 5(4)² - p(4) + 24 = 0

➡ 64 - 80 - 4p + 24 = 0

➡ 8 - 4p = 0

➡ -4p = -8

➡ p = -8/-4

➡ p = 2

VERIFICATION :-

= 64 - 80 - 4(2) + 24

= -16 - 8 + 24

= -24 + 24

= 0

hence, the value of p = 2

Answer 2

$\textbf{\underline{\underline{According\:to\:the\:Question}}}$

x - 4 = 0

x = 4

★Substitute the value of x we get :-

= x^3 - 5x^2 - px + 24

⇒(4)³ - 5(4)² - p(4) + 24 = 0

⇒64 - 80 - 4p + 24 = 0

⇒8 - 4p = 0

⇒-4p = -8

$\tt{\rightarrow p = \dfrac{-8}{-4}}$

 = 2

Value of p = 2