Factor Theorem & Rational Root Theorem

Objective:

SWBAT find zeros of a polynomial by using Rational Root Theorem

The Factor Theorem:

For a polynomial P(x), x – a is a factor iff P(a) = 0

iff “if and only if” It means that a theorem and its converse

are true

If P(x) = x3 – 5x2 + 2x + 8, determine whether x – 4 is a factor. 4 1 -5 2 8

4 -4 -8

1 -1 -2 0

2 3 24 2 82 5x x x xx x

remainder is 0, therefore yes

other factor

Terminology:

Solutions (or roots) of polynomial equations

Zeros of polynomial functions “r is a zero of the function f if f(r) = 0” zeros of functions are the x values of the points

where the graph of the function crosses the x-axis

(x-intercepts where y = 0)

Ex 1: A polynomial function and one of its zeros are given, find the remaining zeros:3 2( ) 3 4 12; 2P x x x x

2 1 3 -4 -12

2 10 12

1 5 6 0

2 5 6 0

2 3 0

2, 3

x x

x x

x

Ex 2: A polynomial function and one of its zeros are given, find the remaining zeros:3( ) 7 6; 3P x x x

-3 1 0 -7 6

-3 9 -6

1 -3 2 0

2 3 2 0

1 2 0

1, 2

x x

x x

x

Rational Root Theorem:

Suppose that a polynomial equation with integral coefficients has the root p/q , where p and q are relatively prime integers. Then p must be a factor of the constant term of the polynomial and q must be a factor of the coefficient of the highest degree term.

(useful when solving higher degree polynomial equations)

Solve using the Rational Root Theorem: 4x2 + 3x – 1 = 0 (any rational root must have a numerator

that is a factor of -1 and a denominator

that is a factor of 4)

factors of -1: ±1

factors of 4: ±1,2,4possible rational roots: (now use synthetic division

to find rational roots)

1 11, ,

2 4

1 4 3 -1

4 7

4 7 6 no

-1 4 3 -1

-4 1

4 -1 0 !yes

4 1 0

4 1

1

4

x

x

x

11,

4x

(note: not all possible rational roots are zeros!)

Ex 3: Solve using the Rational Root Theorem:3 22 13 10 0x x x

1 1 2 -13 10

1 3 -10

1 3 -10 0 !yes

2 3 10 0

5 2 0

5, 2

x x

x x

x

5,1, 2x

1, 2, 5,10possible rational roots:

Ex 4: Solve using the Rational Root Theorem: 3 24 4 0x x x

possible rational roots: 1, 2, 4

1 1 -4 -1 4

1 -3 -4

1 -3 -4 0 !yes

2 3 4 0

4 1 0

1, 4

x x

x x

x

1,1, 4x

Ex 5: Solve using the Rational Root Theorem: 3 23 5 4 4 0x x x

possible rational roots:1 2 4

1, 2, 4, , ,3 3 3

-1 3 -5 -4 4

-3 8

3 -8 -4

-4

0 !yes

23 8 4 0

3 2 2 0

2, 2

3

x x

x x

x

21, , 2

3x

To find other roots can use synthetic divisionusing other possible roots on these coefficients.(or factor and solve the quadratic equation)

2 3 -8 4 3 2 0

6 -4 3 2

3 -2 0

x

x

2

3x