Chapter 15

Solving Polynomials with Powers Three and Higher

A polynomial is a smooth curve that goes on and on forever, using input variables going from negative infinity to positive infinity. To solve a polynomial means to set the equation equal to 0 and determine which, if any, numbers create a true statement. Any numbers satisfying this equation give you important information: They tell you where the graph of the polynomial crosses or touches the x-axis.

The Problems You'll Work On

Solving polynomials in this chapter requires the following techniques:

  • Counting the number of possible real roots/zeros, using Descartes's Rule of Signs
  • Making a list of the possible rational roots/zeros, using the Rational Root Theorem
  • Putting Descartes's Rule of Signs and the Rational Root Theorem together to find roots
  • Applying the Factor Theorem
  • Solving polynomial equations by factoring
  • Applying synthetic division

What to Watch Out For

As you probably know, you can come up with a different answer to a math problem by simply confusing or forgetting one step; here are some things to watch out for:

  • Confusing real roots with rational roots; rational roots are real, but real roots aren't necessarily rational
  • Being sure to list all the possible divisors of a number, not missing multiples
  • Remembering to change the sign of the numerical part of the divisor when using synthetic division
  • Taking roots with multiplicity of more than one into account when looking for factors

Applying Descartes's Rule of Signs to Count Real Roots

656–659 Count the possible number of positive and negative real roots of the equation.

656. x4 − 3x3 + 2x2 − 4x − 9 = 0

657. x5 − x3 + 4x + 1 = 0

658. 5x4 − 3x3 + 6x − 2 = 0

659. x6 + x4 − x3 + 6x2 − x + 9 = 0

Applying the Rational Root Theorem to List Roots

660–663 List all the possible rational roots for each polynomial equation.

660. x5 + x4 − 4x3 − 2x2 − 4x + 8 = 0

661. 5x4 − 3x2 + 6x − 6 = 0

662. 2x5 − 5x4 + 2x3 − 3x2 + 4 = 0

663. 6x4 − 3x3 + 2x2 + 5x + 3 = 0

Determining Whether Numbers Are Roots

664–667 Check to see which of the given values are roots of the equation.

664. Given x3 − 3x2 + 2x + 24 = 0, check to see whether 2, −2, 3, or 4 is a root.

665. Given x4 − 5x3 + 3x2 + 8x + 3 = 0, check to see whether 1, −1, 3, or −3 is a root.

666. Given x5 − 4x4 − 3x3 + 4x + 2 = 0, determine whether 1, −1, 2, or −2 is a root.

667. Given x6 − x5 + x3 − 2x + 1 = 0, determine whether 1 or −1 are roots.

Solving for the Roots of Polynomials

668–685 Solve for all real roots.

668. x3 + 3x2 − 4x − 12 = 0

669. x3 − x2 − 25x + 25 = 0

670. x3 + 4x2 + x − 6 = 0

671. x3 − x2 − 26x − 24 = 0

672. x4 − 81 = 0

673. x6 − 64 = 0

674. x3 + 7x2 + 8x − 16 = 0

675. x3 − 9x2 + 24x − 20 = 0

676. x4 − 37x2 + 36 = 0

677. x4 − 73x2 + 576 = 0

678. x4 − 4x3 − 3x2 + 10x + 8 = 0

679. x5 − x4 − 22x3 − 44x2 − 24x = 0

680. 4x3 − 9x2 − 4x + 9 = 0

681. 4x3 + 12x2 − 9x − 27 = 0

682. 2x5 + 5x4 − 5x3 − 5x2 + 3x = 0

683. 3x4 − 5x3 − 77x2 + 125x + 50 = 0

684. 8x4 − 30x3 − 51x2 + 263x − 210 = 0

685. 5x5 − 6x4 − 14x3 + 28x2 − 15x + 2 = 0

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