Question

A) list all possible rational rootsB) use synthetic division to test several possible rational roots in order to identify one actual root

Answer 1

Part A

The possible rational roots of a polynomial are all the fractions:

$\frac{p}{q}$

where p is a factor of the constant term and q is a factor of the leading coefficient (the constant multiplying the higher power or x).

Thus, for the equation:

$x^4+2x^3-10x^2-18x+9=0$

we have:

• constant term: ,9

,

• factors of 9:, ±1, ±3, ±9

• leading coefficient: ,1

,

• factors of 1: ,±1

Therefore, the possible rational roots of this polynomial are:

$\begin{gathered} \pm\frac{1}{1},\pm\frac{3}{1},\pm\frac{9}{1} \\ \\ -9,-3,-1,1,3,9 \end{gathered}$

Part B

The remainder from the division of a polynomial by (x-a), if a is a root, should be zero. Then, we need to test the possible rational roots using long division until we find a remainder equal to 0.

For a = 3, we obtain:

x³ + 5x² + 5x -3

x - 3 ) x⁴+2x³ -10x² -18x +9

x⁴ -3x³

5x³ -10x² -18x + 9

5x³ -15x²

5x² -18x + 9

5x² -15x

-3x + 9

-3x + 9

0

Thus, since the remainder for the above division is 0, 3 is a root of the polynomial.