HOW TO FIND THE CUBIC POLYNOMIAL WITH GIVEN THREE ZEROES

Question 1 :

Find a polynomial p of degree 3 such that −1, 2, and 3 are zeros of p and p(0) = 1.

Solution :

The zeroes of the polynomial are -1, 2 and 3.

x  =  -1, x  =  2 and x  =  3

From these values, we may find the factors.

The factors are (x + 1) (x - 2) (x - 3)

The required cubic polynomial will be 

p(x)  =  k(x + 1) (x - 2) (x - 3)

p(x)  =  k(x + 1) (x² - 5x + 6)

p(x)  =  k(x³ - 5x² + 6x + x² - 5x + 6)

p(x)  =  k(x³ - 4x² + x + 6)

Given that p(0)  =  1

p(0)  =  k(0³ - 4(0)² + 0 + 6)

1  =  6k

k  =  1/6  =  1/6

p(x)  =  (1/6)(x³ - 4x² + x + 6)

Hence the required polynomial is (1/6)(x³ - 4x² + x + 6).

Question 2 :

Find a polynomial p of degree 3 such that −2, −1, and 4 are zeros of p and p(1) = 2.

Solution :

The zeroes of the polynomial are -2, -1 and 4

x  =  -2, x  =  -1 and x  =  4

From these values, we may find the factors.

The factors are (x + 2) (x + 1) (x - 4)

The required cubic polynomial will be 

p(x)  =  k(x + 2) (x + 1) (x - 4)

p(x)  =  k(x² + 3x + 2) (x - 4)

p(x)  =  k(x³ - 4x² + 3x² - 12x + 2x - 8) 

p(x)  =  k(x³ - x² - 10x - 8)

Given that p(1)  =  2

p(1)  =  k [(1)³ - 1² - 10(1) - 8]

2  =  k (-18)

k  =  -1/9

p(x)  =  (-1/9)(x³ - x² - 10x - 8)

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