Rational Zeros Theorem Calculator

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 leading coefficient is $$$1$$$ or $$$-1$$$). Steps are available.

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Your Input

Find the rational zeros of $$$2 x^{4} + x^{3} - 15 x^{2} - 7 x + 7$$$.

Solution

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 $$$0$$$.

  • Check $$$1$$$: divide $$$2 x^{4} + x^{3} - 15 x^{2} - 7 x + 7$$$ by $$$x - 1$$$.

    The quotient is $$$2 x^{3} + 3 x^{2} - 12 x - 19$$$ and the remainder is $$$-12$$$ (for steps, see synthetic division calculator).

  • Check $$$-1$$$: divide $$$2 x^{4} + x^{3} - 15 x^{2} - 7 x + 7$$$ by $$$x - \left(-1\right) = x + 1$$$.

    The quotient is $$$2 x^{3} - x^{2} - 14 x + 7$$$ and the remainder is $$$0$$$ (for steps, see synthetic division calculator).

  • Hence, $$$-1$$$ is a root.

  • Check $$$\frac{1}{2}$$$: divide $$$2 x^{4} + x^{3} - 15 x^{2} - 7 x + 7$$$ by $$$x - \frac{1}{2}$$$.

    The quotient is $$$2 x^{3} + 2 x^{2} - 14 x - 14$$$ and the remainder is $$$0$$$ (for steps, see synthetic division calculator).

  • Hence, $$$\frac{1}{2}$$$ is a root.

  • Check $$$- \frac{1}{2}$$$: divide $$$2 x^{4} + x^{3} - 15 x^{2} - 7 x + 7$$$ by $$$x - \left(- \frac{1}{2}\right) = x + \frac{1}{2}$$$.

    The quotient is $$$2 x^{3} - 15 x + \frac{1}{2}$$$ and the remainder is $$$\frac{27}{4}$$$ (for steps, see synthetic division calculator).

  • Check $$$7$$$: divide $$$2 x^{4} + x^{3} - 15 x^{2} - 7 x + 7$$$ by $$$x - 7$$$.

    The quotient is $$$2 x^{3} + 15 x^{2} + 90 x + 623$$$ and the remainder is $$$4368$$$ (for steps, see synthetic division calculator).

  • Check $$$-7$$$: divide $$$2 x^{4} + x^{3} - 15 x^{2} - 7 x + 7$$$ by $$$x - \left(-7\right) = x + 7$$$.

    The quotient is $$$2 x^{3} - 13 x^{2} + 76 x - 539$$$ and the remainder is $$$3780$$$ (for steps, see synthetic division calculator).

  • Check $$$\frac{7}{2}$$$: divide $$$2 x^{4} + x^{3} - 15 x^{2} - 7 x + 7$$$ by $$$x - \frac{7}{2}$$$.

    The quotient is $$$2 x^{3} + 8 x^{2} + 13 x + \frac{77}{2}$$$ and the remainder is $$$\frac{567}{4}$$$ (for steps, see synthetic division calculator).

  • Check $$$- \frac{7}{2}$$$: divide $$$2 x^{4} + x^{3} - 15 x^{2} - 7 x + 7$$$ by $$$x - \left(- \frac{7}{2}\right) = x + \frac{7}{2}$$$.

    The quotient is $$$2 x^{3} - 6 x^{2} + 6 x - 28$$$ and the remainder is $$$105$$$ (for steps, see synthetic division calculator).

Answer

Possible rational roots: $$$\pm 1$$$, $$$\pm \frac{1}{2}$$$, $$$\pm 7$$$, $$$\pm \frac{7}{2}$$$A.

Actual rational roots: $$$-1$$$, $$$\frac{1}{2}$$$A.