Question

f(x) is a cubic function whose roots are 2, 3, and 4. Find f(x) and also find the value of f(2).
Please give an explanation so I know how to do it next time 🙂 Thank you!

Step-by-step explanation:

2. If you successfully guess one root of the cubic equation, you can factorize the cubic polynomial using the Factor Theorem and then solve the resulting quadratic equation easily.

33−22−9−4=0
3
x
3

2
x
2

9
x

4
=
0

On inspection, =−1
x
=

1
is found to satisfy the equation. Now the cubic polynomial can be factorized.

33−22−9−4=0
3
x
3

2
x
2

9
x

4
=
0

⟹(+1)(32−5−4)=0

(
x
+
1
)
(
3
x
2

5
x

4
)
=
0

One root is −1

1
. The rest at the roots of the following quadratic equation.

32−5−4=0
3
x
2

5
x

4
=
0

⟹=−(−5)±(−5)2−4×3×(−4)‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾√2×3

x
=

(

5
)
±
(

5
)
2

4
×
3
×
(

4
)
2
×
3

⟹=5±25+48‾‾‾‾‾‾‾‾√6

x
=
5
±
25
+
48
6

⟹=5±73‾‾‾√6

x
=
5
±
73
6

⟹∈{−1,5−73‾‾‾√6,5+73‾‾‾√6}

x

{

1
,
5

73
6
,
5
+
73
6
}

Some cubic equations to be solved by students are deliberately made to have one simple root which can be guessed.