The calculator will find all possible rational roots of the polynomial using the rational zeros theorem. After this, it will decide which possible roots are actually the roots. This is a more general case of the integer (integral) root theorem (when the leading coefficient is $$1$$$or $$-1$$$). Steps are available.

You are watching: What are the possible rational zeros of f(x) = x4 + 2x3 − 3x2 − 4x + 12?

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

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

The trailing coefficient (the coefficient of the constant term) is $$7$$$. Find its factors (with the plus sign and the minus sign): $$\pm 1$$$, $$\pm 7$$$. These are the possible values for $$p$$$.

The leading coefficient (the coefficient of the term with the highest degree) is $$2$$$. Find its factors (with the plus sign and the minus sign): $$\pm 1$$$, $$\pm 2$$$. These are the possible values for $$q$$$.

Find all possible values of $$\frac{p}{q}$$$: $$\pm \frac{1}{1}$$$, $$\pm \frac{1}{2}$$$, $$\pm \frac{7}{1}$$$, $$\pm \frac{7}{2}$$$. Simplify and remove the duplicates (if any). These are the possible rational roots: $$\pm 1$$$, $$\pm \frac{1}{2}$$$, $$\pm 7$$$, $$\pm \frac{7}{2}$$$. Next, check the possible roots: if $$a$$$ is a root of the polynomial $$P{\left(x \right)}$$$, the remainder from the division of $$P{\left(x \right)}$$$ by $$x - a$$$should equal $$0$$$ (according to the remainder theorem, this means that $$P{\left(a \right)} = 0$$$). Check $$1$$$: divide $$2 x^{4} + x^{3} - 15 x^{2} - 7 x + 7$$$by $$x - 1$$$.

$$P{\left(1 \right)} = -12$$$; thus, the remainder is $$-12$$$.

Check $$-1$$$: divide $$2 x^{4} + x^{3} - 15 x^{2} - 7 x + 7$$$ by $$x - \left(-1\right) = x + 1$$$. $$P{\left(-1 \right)} = 0$$$; thus, the remainder is $$0$$$. Hence, $$-1$$$ is a root.

Check $$\frac{1}{2}$$$: divide $$2 x^{4} + x^{3} - 15 x^{2} - 7 x + 7$$$ by $$x - \frac{1}{2}$$$. $$P{\left(\frac{1}{2} \right)} = 0$$$; thus, the remainder is $$0$$$. Hence, $$\frac{1}{2}$$$ is a root.

Check $$- \frac{1}{2}$$$: divide $$2 x^{4} + x^{3} - 15 x^{2} - 7 x + 7$$$ by $$x - \left(- \frac{1}{2}\right) = x + \frac{1}{2}$$$. $$P{\left(- \frac{1}{2} \right)} = \frac{27}{4}$$$; thus, the remainder is $$\frac{27}{4}$$$. Check $$7$$$: divide $$2 x^{4} + x^{3} - 15 x^{2} - 7 x + 7$$$by $$x - 7$$$.

$$P{\left(7 \right)} = 4368$$$; thus, the remainder is $$4368$$$.

Check $$-7$$$: divide $$2 x^{4} + x^{3} - 15 x^{2} - 7 x + 7$$$ by $$x - \left(-7\right) = x + 7$$$. $$P{\left(-7 \right)} = 3780$$$; thus, the remainder is $$3780$$$. Check $$\frac{7}{2}$$$: divide $$2 x^{4} + x^{3} - 15 x^{2} - 7 x + 7$$$by $$x - \frac{7}{2}$$$.

$$P{\left(\frac{7}{2} \right)} = \frac{567}{4}$$$; thus, the remainder is $$\frac{567}{4}$$$.

Check $$- \frac{7}{2}$$$: divide $$2 x^{4} + x^{3} - 15 x^{2} - 7 x + 7$$$ by $$x - \left(- \frac{7}{2}\right) = x + \frac{7}{2}$$$. $$P{\left(- \frac{7}{2} \right)} = 105$$$; thus, the remainder is $$105$$$. See more: Streamlabs Obs Failed To Connect To Streaming Server, How Do I Fix Obs Failed To Connect To The Server Possible rational roots: $$\pm 1$$$, $$\pm \frac{1}{2}$$$, $$\pm 7$$$, $$\pm \frac{7}{2}$$\$A.