Finite Difference

Algebra Level 4

Let P(x) be a cubic polynomial of the form $a{ x }^{ 3 }+b{ x }^{ 2 }+cx+d$

IF: P(1) = 12 P(2) = 44 P(3) = 140 P(4) = 330

Then what does P(5) equal?

Hint: making a triangle of differences may help

2 solutions

Sai Ram
Jul 23, 2015

From the equation , we know that

$P(1)=a+b+c+d=12$

$P(2)=8a+4b+2c+d=44$

$P(3)=27a+9b+3c+d=140$

$P(4)=64a+16d+4c+d=330$

By solving these equations , we get the following values :

$a=5$

$b=2$

$c=-9$

$d=14$ .

Therefore $P(5)=(5(5)^3)+(2(5)^2)-9(5)+14$ = $644$

Arjen Vreugdenhil
Sep 29, 2015

First differences: 32, 96, 190.

Second differences: 64, 94.

Third differences: 30.

For a cubic polynomial, the third differences will be constant. Use this to generate the next value.

Third differences: 30, 30

Second differences: 64, 94, 124

First differences: 32, 96, 190, 314

Values of P: 12, 44, 140, 330, 644 .