Question

Use the Rational Zero Theorem to find the rational zeros of f(x) = x³ - 5x² + 2x + 1.

Answer 1

Step-by-step explanation:

The polynomial is given below as

`f(x)=x^3-5x^2+2x+1`

Since all coefficients are integers, we can apply the rational zeros theorem.

The trailing coefficient (the coefficient of the constant term) is

`=1`

Find its factors (with the plus sign and the minus sign):

The factors will be

`\pm1`

These are the possible values for p

The leading coefficient (the coefficient of the term with the highest degree) is

`=1`

Find its factors (with the plus sign and the minus sign):

The factors will be

`\pm1`

These are the possible values for q

Find all possible values of

`(p)/(q)=\pm1`

Next, check the possible roots: if aa is a root of the polynomial the remainder from the division will be zero

Check 1:

`\begin{gathered} f(x)=x^(3)-5x^(2)+2x+1 \\ f(1)=(1)^3-5(1)^2_+2(1)+1 \\ f(1)=1-5+2+1 \\ f(1)=-1 \end{gathered}`

Check 2:

`\begin{gathered} f(x)=x^(3)-5x^(2)+2x+1 \\ f(-1)=(-1)^3-5(-1)^2+2(-1)+1 \\ f(-1)=-1-5-2+1 \\ f(-1)=-7 \end{gathered}`

Hence,

The possible rational roots are

`\pm1`

Hence,

This polynomial has no rational zeroes