Question

List all the rational roots of f(x) = 24x3 - 10x2 - 7x +2.

Answer 1

If you have to, use the rational roots theorem, if not, use a calculator to find a rational root first (this will be faster)

rational roots theorem possible zeroes will be (i'm just listing all the factors of the constant, 2, over the factors of the leading coefficient, 24)

plus or minus 2/24, 2/12, 2/8, 2/4, 2/3, 2/2, 2/1, 1/24, 1/12, 1/8, 1/4, 1/3, 1/2, 1/1

which is why a calculator will save time.

Answer 2

You can try looking for something with 3 factors:
$$
(ax+b)(cx+d)(ex+f)= 24x^3 - 10x^2 - 7x +2
$$
You can then equate terms:
$$
\large{
abcx^3=24x^3\\
(ac+ce+ae)x^2=-10x^2\\
(ad+cf+af)x=-7x\\
bdf=2\\
}
$$
Solving these, you get
$$
a=2,b=1,c=3,d=-2,e=4,f=-1
$$
Therefore,
$$
(2 x+1) (3 x-2) (4 x-1) = 0\implies x = -1/2,x = 1/4,x = 2/3
$$

Answer 3

That looks like a reasonable solution but a whole lot of work... if you find one rational zero, you can just divide it out.

Answer 4

thank you both!

Answer 5

YW - One other solution that can help is using a spreadsheet (with some fine-tuning around x=0.7):

x f(x)
-0.7 -6.232
-0.65 -4.266
-0.6 -2.584
-0.55 -1.168
\(\bf\color{red}{-0.5}\) 3.9968E-15
-0.45 0.938
-0.4 1.664
-0.35 2.196
-0.3 2.552
-0.25 2.75
-0.2 2.808
-0.15 2.744
-0.1 2.576
-0.05 2.322
9.71445E-17 2
0.05 1.628
0.1 1.224
0.15 0.806
0.2 0.392
\(\bf\color{red}{0.25}\) 0
0.3 -0.352
0.35 -0.646
0.4 -0.864
0.45 -0.988
0.5 -1
0.55 -0.882
0.6 -0.616
0.65 -0.184
\(\bf\color{red}{0.7}\) 0.432
0.75 1.25
0.8 2.288
0.85 3.564
0.9 5.096
0.95 6.902
1 9
1.05 11.408
1.1 14.144