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!
Question
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!
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.