The Remainder and Factor The Remainder and Factor TheoremsTheorems

Check for Understanding 2.3 – Factor polynomials using a variety of methods including the factor theorem, synthetic division, long division, sums and differences of cubes, and grouping.

The Remainder TheoremIf a polynomial f(x) is divided by (x – a), the remainder is the constant f(a), and

f(x) = q(x) ∙ (x – a) + f(a)

where q(x) is a polynomial with degreeone less than the degree of f(x).

Dividend equals quotient times divisor plus remainder.

The Remainder TheoremFind f(3) for the following polynomial function.

f(x) = 5x2 – 4x + 3

f(3) = 5(3)2 – 4(3) + 3

f(3) = 5 ∙ 9 – 12 + 3

f(3) = 45 – 12 + 3

f(3) = 36

The Remainder TheoremNow divide the same polynomial by (x – 3).

5x2 – 4x + 3

3 5 –4 3

5 3611

3315

The Remainder Theorem5x2 – 4x + 3

3 5 –4 3 15 33 5 11 36

f(x) = 5x2 – 4x + 3

f(3) = 5(3)2 – 4(3) + 3

f(3) = 5 ∙ 9 – 12 + 3

f(3) = 45 – 12 + 3

f(3) = 36

Notice that the value obtained when evaluating the function at f(3) and the value of the remainder when dividing the polynomial by x – 3 are the same.

Dividend equals quotient times divisor plus remainder.

5x2 – 4x + 3 = (5x2 + 11x) ∙ (x – 3) + 36

The Remainder TheoremUse synthetic substitution to find g(4) for the following function.

f(x) = 5x4 – 13x3 – 14x2 – 47x + 1

4 5 –13 –14 –47 1 20 28 56 36 5 7 14 9 37

The Remainder TheoremSynthetic Substitution – using syntheticdivision to evaluate a function

This is especially helpful for polynomials withdegree greater than 2.

The Remainder TheoremUse synthetic substitution to find g(–2) for the following function.

f(x) = 5x4 – 13x3 – 14x2 – 47x + 1

–2 5 –13 –14 –47 1 –10 46 –64 222 5 –23 32 –111 223

The Remainder TheoremUse synthetic substitution to find c(4) for the following function.

c(x) = 2x4 – 4x3 – 7x2 – 13x – 10

4 2 –4 –7 –13 –10 8 16 36 92 2 4 9 23 82

Time for Class work Time for Class work

Evaluate each function at the given value.

The Factor TheoremThe binomial (x – a) is a factor of thepolynomial f(x) if and only if f(a) = 0.

The Factor TheoremWhen a polynomial is divided by one of itsbinomial factors, the quotient is called adepressed polynomial.

If the remainder (last number in a depressedpolynomial) is zero, that means f(#) = 0. Thisalso means that the divisor resulting in a remainder of zero is a factor of the polynomial.

The Factor Theoremx3 + 4x2 – 15x – 18 x – 3

3 1 4 –15 –18 3 21 18 1 7 6 0

Since the remainder is zero, (x – 3) is a factor of x3 + 4x2 – 15x – 18.

This also allows us to find the remaining factors of the polynomial by factoring the depressed polynomial.

The Factor Theoremx3 + 4x2 – 15x – 18 x – 3

3 1 4 –15 –18 3 21 18 1 7 6 0

x2 + 7x + 6

(x + 6)(x + 1)

The factors of

x3 + 4x2 – 15x – 18

are

(x – 3)(x + 6)(x + 1).

The Factor Theorem

(x – 3)(x + 6)(x + 1).

Compare the factors of the polynomials

to the zeros as seenon the graph of

x3 + 4x2 – 15x – 18.

The Factor TheoremGiven a polynomial and one of its factors, find the remainingfactors of the polynomial. Some factors may not be binomials.

1. x3 – 11x2 + 14x + 80 x – 8

2. 2x3 + 7x2 – 33x – 18 x + 6

(x – 8)(x – 5)(x + 2)

(x + 6)(2x + 1)(x – 3)

Using the Factor Theorem, determine if f(x) is a factor of p(x)

The Factor Theorem

Time for Class work Time for Class work

Using the Factor Theorem, factor fully each of the following Using the Factor Theorem, factor fully each of the following polynomials: polynomials:

Using the Factor Theorem, determine if f(x) is a factor of p(x)

The Rational Zero Theorem

The Rational Zero Theorem gives a list of possible rational zeros of a polynomial function. Equivalently, the theorem gives all possible rational roots of a polynomial equation. Not every number in the list will be a zero of the function, but every rational zero of the polynomial function will appear somewhere in the list.

The Rational Zero Theorem

If f (x) anxn an-1x

n-1 … a1x a0 has integer coefficients and

(where is reduced) is a rational zero, then p is a factor of the

constant term a0 and q is a factor of the leading coefficient an.

The Rational Zero Theorem

If f (x) anxn an-1x

n-1 … a1x a0 has integer coefficients and

(where is reduced) is a rational zero, then p is a factor of the

constant term a0 and q is a factor of the leading coefficient an.

and pq

pq

3 2

List the potential rational zeros of3 8 7 12f x x x x

: 1, 2, 3, 4, 6, 12p : 1, 3q

1 2 4: 1, 2, 3, 4, 6, 12, , ,

3 3 3

p

q

Factors of the constant

Factors of the leading coefficient

EXAMPLE: Using the Rational Zero TheoremList all possible rational zeros of f (x) 15x3 14x2 3x – 2.

Solution The constant term is 2 and the leading coefficient is 15.

1 2 1 2 1 25 53 3 15 15

Factors of the constant term, 2Possible rational zerosFactors of the leading coefficient, 15

1, 21, 3, 5, 15

1, 2, , , , , ,

Divide 1

and 2 by 1.

Divide 1

and 2 by 3.

Divide 1

and 2 by 5.

Divide 1

and 2 by 15.

There are 16 possible rational zeros. The actual solution set to f (x) 15x3 14x2 3x – 2 = 0 is {-1, 1/3, 2/5}, which contains 3 of the 16 possible solutions.

22

The Rational Zero Test

factors of the constant term

factors oPossible

f the learati

dingonal

coefzeros =

ficient

Example

Find all potential rational zeros of

4 3 2( ) 2 17 35 9 45f x x x x x

p

q

Solution

23

The Rational Zero Test (continued)

Example

Use the Rational Zero Test to find ALL rational zeros of

4 2( ) 4 17 4f x x x