Use the Rational Roots Test to Find All Possible Roots f(x)=x^5+3x^4*8x^3+14x^2+16x+8

$f (x) = x^{5} + 3 x^{4} \cdot 8 x^{3} + 14 x^{2} + 16 x + 8$

Step 1

Simplify.

Simplify each term.

Multiply $x^{4}$ by $x^{3}$ by adding the exponents.

Move $x^{3}$ .

$x^{5} + 3 (x^{3} x^{4}) \cdot 8 + 14 x^{2} + 16 x + 8$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{5} + 3 x^{3 + 4} \cdot 8 + 14 x^{2} + 16 x + 8$

Add $3$ and $4$ .

$x^{5} + 3 x^{7} \cdot 8 + 14 x^{2} + 16 x + 8$

Multiply $8$ by $3$ .

$x^{5} + 24 x^{7} + 14 x^{2} + 16 x + 8$

Reorder $x^{5}$ and $24 x^{7}$ .

$24 x^{7} + x^{5} + 14 x^{2} + 16 x + 8$

Step 2

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 2, \pm 4, \pm 8$

$q = \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24$

Step 3

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{4}, \pm \frac{1}{6}, \pm \frac{1}{8}, \pm \frac{1}{12}, \pm \frac{1}{24}, \pm 2, \pm \frac{2}{3}, \pm 4, \pm \frac{4}{3}, \pm 8, \pm \frac{8}{3}$

Use the Rational Roots Test to Find All Possible Roots f(x)=x^5+3x^4*8x^3+14x^2+16x+8

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