[PDF] The Remainder and Factor Theorems PDF

[PDF] 6 The factor theorem

If f(x) is a polynomial and f(a) = 0, then (x–a) is a factor of f(x) Proof of the factor theorem Let's start with an example Consider 4 8 5

[PDF] 32 The Factor Theorem and The Remainder Theorem

The Factor Theorem: Suppose p is a nonzero polynomial It is important to note that it works only for these kinds of divisors 5 Also Example 3 2 1

[PDF] AMSG11Remainder and Factor Theorempdf

Now consider another example of a cubic polynomial divided by a linear divisor From the above example, we can deduce that: 2 ? 3 + 4 + 5 =

[PDF] 23 Factor and remainder theorems

1 Factorise polynomial expressions 2 3 2 Divide a polynomial by a linear or quadratic factor 2 3 3 Apply the remainder theorem 2 3 4 Apply the factor theorem

[PDF] The Factor Theorem

For example, the zeros of p are –3, 1, and 5, and the factors of p(x) are x + 3, x - 1, and x - 5 The following theorem generalizes this relationship

[PDF] Factor Theorem Examples And Solutions

2 Remainder and Factor Theorems Interactive Mathematics Factor theorem state with proof examples and solutions factorise the Polynomials Maths Mutt

[PDF] 51 The Remainder and Factor Theorems; Synthetic Division

Example 8: 7 5 4 3)( 2 3 + ? + = x x x xf Find )4( ? f using (a) synthetic division (b) the Remainder Theorem Example 9: Solve the equation

[PDF] 428 - The Factor Theorem - Scoilnet

4 2 - Algebra - Solving Equations 4 2 8 - The Factor Theorem Higher Level ONLY 1 / 5 Example 1 Q Suppose f (x)=5x3 - 14x2 + 12x - 3

[PDF] 22 - The Factor Theorem

24 fév 2015 · Use long division to determine the other factors Page 6 6 February 24, 2015 Example Five Factor fully

[PDF] 53 Factor & Remainder Theorem

PROBLEM You Have the Right to the Remainder Theorem Chapter 5 Polynomial Expressions and Equations example are correct? - Long Division

[PDF] 6 The factor theorem

linear factors corresponding to the zeros x=1,2 and 4 That is, Proof of the factor theorem Let's start with an example Consider 4 8 5 2 16 4 18 4 32 8 36

[PDF] 51 The Remainder and Factor Theorems; Synthetic Division

xf Example 8: 7 5 4 3)( 2 3 + − + = x x x xf Find )4( − f using (a) synthetic division (b) the Remainder Theorem Example 9: Solve the equation 06 11 3

[PDF] 32 The Factor Theorem and The Remainder Theorem

The Remainder Theorem: Suppose p is a polynomial of degree at least 1 Now take the 2 from the divisor times the 6 to get 12, and add it to the -5 to get 7 The next example pulls together all of the concepts discussed in this section

[PDF] Section 34 Factor Theorem and Remainder Theorem

polynomial division Example 1 Divide 14 5 4 2 3

[PDF] The Remainder and Factor Theorems

p Use polynomial division in real-life problems 6 5 5 3 2 2 6 1 1 3 9 Your Notes REMAINDER THEOREM If a polynomial f(x) is divided by x

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101353_6Alg2_6_5.pdf

The Remainder and Factor

Theorems

GoalspDivide polynomials and relate the result to the remainder theorem and the factor theorem. pUse polynomial division in real-life problems. 6.5

VOCABULARY

Polynomial long divisionA method used to divide

polynomials similar to the way you divide numbers

Synthetic divisionA method used to divide a

polynomial by an expression of the form x?k

Your Notes

Divide 4x

4 ?x 2 ?18x?8 by x 2 ?2x?3. Write division in the same format you would use when dividing numbers. Include a 0 as the coefficient of x 3 . x 2 ?2x?3?4?x 4 ???0?x 3 ???1?2?x 2 ???1?8?x???8?

Write the result as follows.

?4x 2 ?8x?3 ? 1 ?? x 2 ?2x?3 4x 4 ?x 2 ?18x?8 ??? x 2 ?2x?3 3x 2 ? x 2 ?8x 3 ? x 2 4x 4 ? x 2

Example 1Using Polynomial Long Division

4x 4 ?8x 3 ?12x 2 ?8x 3 ?13x 2 ?18x ?8x 3 ?16x 2 ?24x 4x 2 ?8x?3 3x 2 ?6x?8 3x 2 ?6x?9 ?1

136Algebra 2 Notetaking Guide Chapter 6

Divide x

3 ?x 2 ?5x?3 by x?2.

Solution

To find the value of k, rewrite the divisor in the form x?k.

Because x?2 ?x?(?2),k? 2.

? x 2 ?x?3 ? 9 ? x ? 2 x 3 ?x 2 ?5x?3 ??? x?2 ?211?53 ?226

1?1?39

Your Notes

REMAINDER THEOREM

If a polynomial f(x) is divided by x?k, the remainder is r?f(k).

Example 2Using Synthetic Division

Factor f(x) ? x

3 ?19x?30 given that f(5) ? 0.

Solution

Because f(5) ?0, you know that x?5is a factor of f(x).

Use synthetic division to find the other factors.

The result gives the coefficients of the quotient. x 3 ?19x?30 ?( x?5 )( x 2 ?5x?6 ) ?( x?5 )( x?2 )( x?3 )

510?19?30

52530

15 6 0

FACTOR THEOREM

A polynomial f(x) has a factor x?kif and only if f(k) ?0.

Example 3Factoring a Polynomial

Lesson 6.5 Algebra 2 Notetaking Guide137

A zero of f(x) ?x

3 ?x 2 ?4x?4 is x??1. Find the other zeros.

Solution

Because f(?1) ?0, you know that x?1is a factor of f(x).

Use synthetic division to find the other factors.

The result gives the coefficients of the quotient. f(x) ?x 3 ?x 2 ?4x?4 ?( x?1)( x 2 ?4) ?( x?1)( x?2)( x?2) By the factor theorem, the zeros of fare ?1,?2, and 2 . ?111?4?4 ?104 10?40

[PDF] The Remainder and Factor Theorems

The Remainder and Factor

Theorems

VOCABULARY

Polynomial long divisionA method used to divide

Synthetic divisionA method used to divide a

Your Notes

Divide 4x

Write the result as follows.

Example 1Using Polynomial Long Division

136Algebra 2 Notetaking Guide  Chapter 6

Divide x

Solution

Because x?2 ?x?(?2),k? 2.

1?1?39

Your Notes

REMAINDER THEOREM

Example 2Using Synthetic Division

Factor f(x) ? x

Solution

Use synthetic division to find the other factors.

510?19?30

15 6 0

FACTOR THEOREM

Example 3Factoring a Polynomial

Lesson 6.5 Algebra 2 Notetaking Guide137

A zero of f(x) ?x

Solution

Use synthetic division to find the other factors.

Your Notes

Example 4Finding Zeros of a Polynomial Function

CheckpointComplete the following exercises.

1.Use long division to divide x

2.Use synthetic division to divide 2x

3.Factor f(x) ?2x

4.A zero of f(x) ?x

Homework

138Algebra 2 Notetaking Guide  Chapter 6

Factor Theorem Documents PDF, PPT , Doc

136Algebra 2 Notetaking Guide Chapter 6

Lesson 6.5 Algebra 2 Notetaking Guide137

138Algebra 2 Notetaking Guide Chapter 6