Question

If x = 2 + √3 the value of x3 + 3x2 – 27 x + 87 is

Answer 1

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${\underline{\bf{Given,}}}$

$\small{\leadsto\sf{x = 2 + \sqrt{3}}}$

$\small{\leadsto\sf{p(x)=x^{3}+3x^{2}-27x+87}}$

${\underline{\bf{To \:find,}}}$

the value of $\small{\sf{p(x)=x^{3} + 3x^{2} - 27x + 87}}$

${\underline{\bf{Solution,}}}$

Putting, x = 2 + √3 in p(x) = x³+ 3x² - 27x + 87

$\dashrightarrow$ (2 + √3)³ + 3(2 + √3)² - 27(2 + √3) + 87

$\dashrightarrow$(2)³ + (√3)³ + 3(2)(√3)(2 + √3) + 3(2 + √3)² - 27(2 + √3) + 87

$\dashrightarrow$4 + 3√3 + 6√3( 2 + √3) + 3{(2)²+ (√3)²+2(2)(√3)} - 54 - 27√3 + 87

$\dashrightarrow$4 + 3√3 + 12√3 + 54 + 3(4 + 3 + 4√3) - 27√3 + 33

$\dashrightarrow$4 + 3√3 + 12√3 + 54 + 12 + 9 + 12√3 - 27√3 + 33

$\dashrightarrow$3√3 + 12√3 + 12√3 - 27√3 + 4 + 54 + 12 + 9 + 33

$\dashrightarrow$ $\cancel{24\sqrt{3}}\cancel{-24\sqrt{3}}$ + 112

$\dashrightarrow$112

$\therefore{\boxed{\sf{\pink{p(x)=x^{3}+3x^{2}-27x+87=112}}}}$

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Identities used:-

• (a + b)³ = a³ + b³ + 3ab(a + b)
• (a + b)² = a² + b² + 2ab

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