Question

Write a polynomial function in standard form with zeros at 1, 2, and 3.

Answer 1

The polynomial whose zeroes are 1, 2 and 3 is given by,
(x-1)(x-2)(x-3) = 0
(x-1)[x²-3x-2x+6]=0
(x-1)[x²-5x+6]=0
x[x²-5x+6] -1[x²-5x+6]=0
x³-5x²+6x-x²+5x-6=0
x³-6x²+11x-6=0

Therefore, the required polynomial is,
x³ - 6x² + 11x -6 = 0

Answer 2

Answer:  The required polynomial in standard form is $x^2-6x^2+11x-6.$

Step-by-step explanation:  We are given to write a polynomial in standard form with zeroes 1, 2 and 3.

We know that

if a polynomial p(x) has zeroes at a, b and c, then its expression can be written as follows :

$p(x)=(x-a)(x-b)(x-c).$

Therefore, if f(x) represents the given polynomial, then we must have

$f(x)\\\\=(x-1)(x-2)(x-3)\\\\=(x^2-3x+2)(x-3)\\\\=x(x^2-3x+2)-3(x^2-3x+2)\\\\=x^3-3x^2+2x-3x^2+9x-6\\\\=x^2-6x^2+11x-6.$

Thus, the required polynomial in standard form is $x^2-6x^2+11x-6.$