Question

Hi, can you help me to solve this exercise, please!!For cach polynomial, LIST all POSSIBLE RATIONAL. ROOTS.•Find all factors of the leading coefficient and constant value of polynonnal.•ANY RATIONAL ROOTS = ‡ (Constant Factor over Leading Coefficient Factor)

Answer 1

The polynomial is given to be:

`x^4+2x^3-8x^2+16x-32`

The Rational Roots Test (also known as Rational Zeros Theorem) allows us to find all possible rational roots of a polynomial. Suppose we have some polynomial P(x) with integer coefficients and a nonzero constant term. If p represents the factors of constant term and q represents the factors of the leading coefficient, the possible rational roots will be:

`\Rightarrow(p)/(q)`

From the polynomial given, the constant term is -32 and the leading coefficient is 1.

Therefore, the factors of the constant will be:

`p=1,2,4,8,16,32`

and the factors of the leading coefficient will be:

`q=1`

Hence, the possible rational roots will be:

`\begin{gathered} \pm(p)/(q)=\pm(1)/(1),\pm(2)/(1),\pm(4)/(1),\pm(8)/(1),\pm(16)/(1),\pm(32)/(1) \\ \therefore \\ \pm(p)/(q)=\pm1,\pm2,\pm4,\pm8,\pm16,\pm32 \end{gathered}`