Chapter 2 · Question 5

Form a quadratic polynomial whose zeroes are $4$ and $-1$ .

Answer

A quadratic with zeroes

\alpha

and

\beta

is

k[x^2-(\alpha+\beta)x+\alpha\beta]

. Taking

k=1

, zeroes

4,-1

give

x^2-3x-4

.

Use sum

=3

and product

=-4

. The polynomial is

x^2-3x-4

.

Here

\alpha=4

and

\beta=-1

. Sum

\alpha+\beta=3

and product

\alpha\beta=-4

. A monic quadratic polynomial is

x^2-(\alpha+\beta)x+\alpha\beta=x^2-3x-4

. Any non-zero multiple of this polynomial has the same zeroes.

A quadratic with zeroes $\alpha$ and $\beta$ is $k[x^2-(\alpha+\beta)x+\alpha\beta]$ .