Question

course
ap precalculus
due mon, oct 27, 2025
2c
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1. ( y = 2 x ^ { 4 } + 11 x ^ { 3 } + x ^ { 2 } - 22 x - 10 )

c. list all of the possible rational roots.
text math set matrix
answer

Explanation:

Step1: Identify the constant term and leading coefficient

For the polynomial \(y = 2x^{4}+11x^{3}+x^{2}-22x - 10\), the constant term \(p=- 10\) and the leading coefficient \(q = 2\).

Step2: Find the factors of \(p\) and \(q\)

The factors of \(p=-10\) are \(\pm1,\pm2,\pm5,\pm10\). The factors of \(q = 2\) are \(\pm1,\pm2\).

Step3: Apply the Rational Root Theorem

The possible rational roots are of the form \(\frac{p}{q}\). So, \(\frac{\pm1}{\pm1},\frac{\pm1}{\pm2},\frac{\pm2}{\pm1},\frac{\pm2}{\pm2},\frac{\pm5}{\pm1},\frac{\pm5}{\pm2},\frac{\pm10}{\pm1},\frac{\pm10}{\pm2}\).
Simplifying these, we get \(\pm1,\pm\frac{1}{2},\pm2,\pm5,\pm\frac{5}{2},\pm10\).

Answer:

The possible rational roots are \(1,-1,\frac{1}{2},-\frac{1}{2},2,-2,5,-5,\frac{5}{2},-\frac{5}{2},10,-10\).