poly poly nomial!!!!!

Algebra Level 3

A cubic polynomial $p(x)$ is such that $p(1)=1$ , $p(2)=2$ , $p(3)=3$ , $p(4)=5$ .Find the value of $p(6)$ .

A polynomial of degree $3$ is called a cubic polynomial.

The answer is 16.

2 solutions

Akshay Mujumdar
Jan 19, 2015

See image below

Advitiya Brijesh
Jul 24, 2014

Let a cubic Polynomial $P(x)=ax^3+bx^2+cx+d$ Then On solving, $a=\frac{1}{6}$ , $b=-1$ , $c=\frac{17}{6}$ , $d=-1$ which gives, $P(6)=16$ :)

Exactly,bro!:D

Anik Mandal - 6 years, 8 months ago

Dont you think that this is a very lengthy method?????? You will put all the conditions, and you will get 4 equations. Then you will have to solve 4 equations simultaneously to find the values of the 4 variables.There is an alternative...... p(1) = 1, p(2) = 2 and p(3) = 3. So, we know that there are 3 real roots of the equation p(x) - x = 0, and these roots are 1, 2 and 3. Since p(x) is a cubic polynomial, p(x) - x is also a cubic polynomial, and hence, it will not have any more roots. Now consider a polynomial g(x) = a[p(x) - x]. The roots of g(x) = 0 are also 1, 2 and 3. If you know all the roots of the equation, you can find the equation. Here, p(x) - x = x^3 - (Sum of roots) x^2 + (Sum of roots taken two at a time)x - (Product of roots). Hence, p(x) - x = x^3 - (1+2+3)x^2 + (1 * 2 + 2 * 3 + 3 * 1)x -(1 * 2 * 3). Hence, p(x) - x = x^3 - 6x^2 + 11x - 6. From g(x) = a[p(x) - x], we know that p(x) = x + (g(x)/a). Since p(4) = 5, we can find out the value of a. a comes out to be 6. Therefore, you can now easily find out p(6), which comes out to be 16. I know that at first sight, this method looks lengthy, but believe me, it is not. You can do this pretty fast!!!!!

Akshay Mujumdar - 6 years, 4 months ago

poly poly nomial!!!!!

The answer is 16.

2 solutions

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