Mathematics

Problem

Consider the monic cubic polynomial

s(x)=x^{3}+ax^{2}+bx+c

. It satisfies the following conditions:

\;

•

(x+1)

is a factor of

s(x)

\;

•

x=2

is a zero of

s(x)

\;

•

s(1)=10

. a. (i) Using the three conditions above, set up three equations in

a

b

and

c

. b. (ii) Solve the system of equations to obtain the values of

a

b

and

c

. c. (iii) Write

s(x)

in its fully factorised form and list all its zeros.

Intuitive explanation

The key is to use the three conditions as a system of equations to find the unknown coefficients $a$ , $b$ , and $c$ . Since $(x+1)$ is a factor, $s(-1)=0$ ; since $x=2$ is a zero, $s(2)=0$ ; and you're given $s(1)=10$ . Plugging each of these into the polynomial gives you three linear equations in three unknowns—a straightforward algebra problem once you recognize that the conditions translate directly into function values.

Intuitive explanation

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