p ( x ) = x 4 + a x 3 + b x 2 + c x + d , where p ( 1 ) = 10 , p ( 2 ) = 20 , p ( 3 ) = 30 , then the value of p ( 12 ) + p ( - 8 ) 10 - 1980 is

Question

$p (x) = x^{4} + a x^{3} + b x^{2} + c x + d$ , where $p (1) = 10$ , $p (2) = 20$ , $p (3) = 30$ , then the value of $\frac{p (12) + p (- 8)}{10} - 1980$ is ________.

Smart Achievers · Accepted Answer

(4)

$Let$ $f (x) = p (x) - 10 x$

$∴$ $f (1) = f (2) = f (3) = 0$

$∴$ $p (x) = (x - 1) (x - 2) (x - 3) (x - λ) + 10 x$

$p (12) = 11 \times 10 \times 9 (12 - λ) + 120$

$p (- 8) = (- 9) \times (- 10) \times (- 11) (- 8 - λ) - 80$

$∴$ $p (12) + p (- 8) = 11 \times 10 \times 9 (12 - λ + 8 + λ) + 40$

$= 11 \times 10 \times 9 \times 20 + 40 = 40 \times 496$

$or$ $\frac{p (12) + p (- 8)}{10} = 1984$

$∴$ $\frac{p (12) + p (- 8)}{10} - 1980 = 4$

Solution

Q.
$p (x) = x^{4} + a x^{3} + b x^{2} + c x + d$ , where $p (1) = 10$ , $p (2) = 20$ , $p (3) = 30$ , then the value of $\frac{p (12) + p (- 8)}{10} - 1980$ is ________.

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Solution

Q. p(x)=x4+ax3+bx2+cx+d, where p(1)=10, p(2)=20, p(3)=30, then the value of p(12)+p(-8)10-1980 is ________.

Ans. (4) Let f(x)=p(x)-10x ∴ f(1)=f(2)=f(3)=0 ∴ p(x)=(x-1)(x-2)(x-3)(x-λ)+10x p(12)=11×10×9(12-λ)+120 p(-8)=(-9)×(-10)×(-11)(-8-λ)-80 ∴ p(12)+p(-8)=11×10×9(12-λ+8+λ)+40 =11×10×9×20+40=40×496 or p(12)+p(-8)10=1984 ∴ p(12)+p(-8)10-1980=4

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Q.
$p (x) = x^{4} + a x^{3} + b x^{2} + c x + d$ , where $p (1) = 10$ , $p (2) = 20$ , $p (3) = 30$ , then the value of $\frac{p (12) + p (- 8)}{10} - 1980$ is ________.