1. 5
Solving Polynomial
Equations

2. Chapter Overview
A goal throughout all of algebra has been to solve equations.
In this chapter, we will develop methods to solve polynomial equations of any degree.

3. 5.1
The Remainder and Factor Theorems; Synthetic Division

4. Objectives Understand the Definition of a Zero of a Polynomial
Use the Remainder Theorem
Use the Factor Theorem
Use Synthetic Division to Divide Polynomials
Use Synthetic Division to Evaluate Polynomials
Use Synthetic Division to Solve Polynomial Equations

5. 1. Understand the Definition of a Zero of a Polynomial

6. Polynomial Equations
A polynomial equation is an equation that can be written in the form P(x) = 0, where
P(x) = anxn + an–1xn–1 + an–2xn–2 + … + a1x + a0
where n is a natural number and the polynomial is of degree n.

7. Zero of a Polynomial Function
A zero of the polynomial P(x) is any number r for which P(r) = 0.

8. Roots and Zeros
In general, the roots of the polynomial equation P(x) = 0 are the zeros of the polynomial P(x) .

9. 2. Use the Remainder Theorem

10. Example 1 Let P(x) = 3x3 – 5x2 + 3x – 10. Find P(1).
Divide P(x) by x – 1.

11. Example 1(a) – Solution
To find P(1), we will substitute 1 for x in the polynomial.

12. Example 1(b) – Solution
To divide P(x) by x – 1, we proceed as follows:

13. Example 1 – Solution
Note that the remainder is equal to P(1).

14. The Remainder Theorem
If P(x) is a polynomial, r is any number, and P(x) is divided by x – r, the remainder is P(r).

15. 3. Use the Factor Theorem

16. The Factor Theorem If P(x) is a polynomial, r is any number, then
If P(r) = 0, then x – r is a factor of P(x).
If x – r is a factor of P(x), then P(r) = 0.

17. Alternate Form of The Factor Theorem
If r is a zero of the polynomial P(x) = 0, then x – r is a factor of P(x).
If x – r is a factor of P(x), then r is a zero of the polynomial.

18. Example 4
Determine whether x + 2 is a factor of
P(x) = x4 – 7x2 – 6x

19. Example 4 – Solution
By the factor theorem, x + 2, or x – (–2), is a factor of P(x) if P(–2) = 0.
So we find the value of P(–2).
Since P(–2) = 0, we know that x – (–2), or x + 2, is a factor of P(x).

20. 4. Use Synthetic Division to Divide Polynomials

21. Synthetic Division
Synthetic division is an easy way to divide higher-degree polynomials by binomials of the form x – r.
And, it is much faster than long division.

22. Example 5
Use synthetic division to divide 10x + 3x4 – 8x by x – 2.

23. Example 5 – Solution
We first write the terms of P(x) in descending powers of x:
3x4 – 8x3 + 10x + 3
We then write the coefficients of the dividend, with its terms in descending powers of x and writing 0 for the coefficient of the missing x2 term, and the 2 from the divisor in the following form:

24. Example 5 – Solution
Then we follow these steps:

25. Example 5 – Solution

26. Example 5 – Solution
Thus,

27. 5. Use Synthetic Division to Evaluate Polynomials

28. Example 6 Use synthetic division to find P(–2) when
P(x) = 5x3 + 3x2 – 21x – 1

29. Example 6 – Solution
Because of the remainder theorem, P(–2) is the remainder when P(x) is divided by x – (–2).
Earlier in the section, we would have found the remainder by using long division.
We now can simplify the work and find the remainder by using synthetic division.

30. Example 6 – Solution
Because the remainder is 13, P(–2) = 13.

31. 6. Use Synthetic Division to Solve Polynomial Equations

32. Example 8 Let P(x) = 3x3 – 5x2 + 3x – 10.
Completely solve the polynomial equation P(x) = 0 given that 2 is one solution.

33. Example 8 – Solution
Since 2 is a solution of the equation P(x) = 0, we know that 2 is a zero of P(x).
We will use synthetic division and divide P(x) by x – 2, obtaining a remainder of 0.
Then we will use the result of the synthetic division to help factor the polynomial and solve the equation.

34. Example 8 – Solution
We use synthetic division to divide P(x) by x – 2.
We then write the quotient and factor it.

35. Example 8 – Solution
Finally, we solve the polynomial equation P(x) = 0.

36. Example 8 – Solution
The solution set is

37. Multiplicity of Zeros
If r is a zero of P(x) that occurs n times, we say that r is a zero of multiplicity n.
For example, has zeros 3, 3, and –5.
Because 3 occurs twice as a zero, we say that 3 is a zero of multiplicity 2.