# Synthetic Division Review

**Review: Synthetic Division
**■ Find (x

^{2}- 5x - 5x

^{3}+ x

^{4}) ÷ (5 + x).

** Factor Theorem
**■ Solve 2x

^{3}- 5x

^{2}+ x + 2 =0 given that 2 is a

zero of f (x) = 2x

^{3}- 5x

^{2}+ x + 2.

Lesson 2.5, page 312

Zeros of Polynomial

Functions

Objective: To find a polynomial with specified zeros, rational zeros , and other zeros. |

**Introduction**

Polynomial | Type of Coefficient |

5x^{3} + 3x^{2} + (2 + 4i) + i |
complex |

real | |

rational | |

5x^{3} + 3x^{2} + 8x – 11 |
integer |

**Rational Zero Theorem
**If the polynomial

has integer coefficients, then every rational _____

of f(x) is of the form

where p is a factor of the ________ coefficient a

_{0}

and q is a factor of the ________ coefficient a

_{n}.

**Rational Root (Zero) Theorem
(in other words)
**If “q” is the leading coefficient and “p” is the

constant term of a polynomial , then the only

possible rational roots are ± factors of “p”

divided by ± factors of “q”. (p / q)

**Rational Root (Zero) Theorem
(in other words)
**Example:

■ To find the POSSIBLE rational roots of f(x), we need the

FACTORS of the leading coefficient (6 for this example)

and the factors of the constant term (4, for this example).

Possible rational roots are

**See Example 1, page 313.
**■
Check Point 1: List all possible rational zeros

of f(x) = x

^{3}+ 2x

^{2}– 5x – 6.

**Another example
**■
Check Point 2: List all possible rational zeros

of f(x) = 4x

^{5}+ 12x

^{4}– x – 3.

**How do we know which possibilities
are really zeros (solutions)?
**■ Use trial and error and ________ division to

see if one of the possible zeros is actually a

zero.

■
Remember: When dividing by x – c, if the

________ is 0 when using synthetic division,

then c is a zero of the polynomial.

■
If c is a zero, then solve the polynomial

resulting from the synthetic division to find the

other zeros.

**See Example 3, page 315.
**■ Check Point 3: Find all zeros of

f(x) = x

^{3}+ 8x

^{2}+ 11x – 20.

**Finding the Rational Zeros of a Polynomial
**1. List all ________ rational zeros of the polynomial

using the Rational Zero Theorem.

2. Use synthetic division on each possible rational zero

and the polynomial until one gives a remainder of

________. This means you have found a zero, as

well as a factor.

3. Write the polynomial as the ________ of this factor

and the quotient.

4. Repeat procedure on the quotient until the quotient is

________

5. Once the quotient is quadratic, factor or use the

quadratic formula to find the remaining real and

imaginary zeros.

**Check Point 4, page 316
**■
Find all zeros of

f(x) = x

^{3}+ x

^{2}- 5x – 2.

List all possible zeros, and use synthetic division

to test and find an actual zero. Then use the

quotient to find the remaining zeros.

■
f(x) = x^{3} – 4x^{2} + 8x - 5

More review -- List all possible zeros. Use synthetic

division to test and find an actual zero. Then use the

resulting quotient to find the remaining zeros. (HW #13)

■
f(x) = x^{3} + 4x^{2} - 3x - 6

How many zeros, not necessarily rational, does

a polynomial with rational coefficients have?

■
An nth degree polynomial has a total of n ________.

Some may be rational, irrational or complex.

■
Because all coefficients are RATIONAL, irrational roots

exist in ________ (both the irrational # and its conjugate).

________ roots also exist in pairs (both the complex # and

its conjugate).

■ If a + bi is a root, a – bi is a root

■ If is root, is a root.

■
NOTE: Sometimes it is helpful to graph the function and

find the x- intercepts (zeros) to narrow down all the

possible zeros.

**See Example 5, page 317.**

■ Check Point 5

■
Solve: x^{4} - 6x^{3} + 22x^{2} - 30x + 13 = 0.

**Fundamental Theorem of Algebra
(page 318)**

■
If f(x) is a polynomial function of degree n,

where n ≥ 1, then the equation f (x) = 0 has at

least one complex zero, real or imaginary.

■
Note: This theorem just guarantees a zero

exists, but does not tell us how to find it.

** Linear Factorization Theorem, pg. 319 **

**Remember…**

■
Complex zeros come in pairs as

complex conjugates: a + bi, a – bi

■
Irrational zeros come in pairs.

**More Practice**

Find a polynomial function, in factored

form, of degree 5 with -1/2 as a zero

with multiplicity 2, 0 as a zero of

multiplicity 1, and 1 as a zero of

multiplicity 2.

**Practice
**Find a polynomial function of degree 3

with 2 and i as zeros.

**See Example 6, page 319.**

■ Check Point 6, page 320

■
Find a third-degree polynomial function f(x)

with real coefficients that has -3 and i as

zeros and such that f(1) = 8.

Solve the given polynomial equation. Use the

Rational Zero Theorem, or graph as an aid to

obtaining the first zero.

■
x^{4} – x^{3} + 2x^{2} – 4x – 8 = 0.

**Extra Example
**Suppose that a polynomial function of degree 4

with rational coefficients has i and

as zeros. Find the other zero(s).

**Extra Example
**Find a polynomial of degree 3 where 4 and 2i

are zeros, and f(-1) = -50.

■
Use the Rational Zero Theorem to list all the

possible zeros for

f(x) = 4x^{5} – 8x^{4} – x + 2.

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