Polynomial amp Synthetic DivisionAlgebra III Sec 23

Objective

Use long division and synthetic division to divide polynomials by other polynomials

Important Vocabulary

Improper ndash the degree of the numerator is greater than or equal to the degree of the denominator

Proper ndash the degree of the numerator is less than the degree of the denominator

Long Division of Polynomials

Dividing polynomials is useful when ________________________

When dividing a polynomial f(x) by another polynomial d(x) if the remainder r(x) = 0 d(x) _____________________ into f(x)

The result of a division problem can be checked by _________________

evaluating functions

divides evenly

multiplication

Example (on your handout)

Divide 3x3 + 4x ndash 2 by x2 + 2x + 1

What can I multiply x2 by to get 3x3

Multiply 3x by x2 + 2x + 1

Subtract

What can I multiply x2 by to get -6x2

Multiply -6 by x2 + 2x + 1

Subtract

Synthetic Divison

Synthetic division is a shortcut for long division of polynomials by divisors of the form x ndash k

Example 4 Use synthetic division to divide 2x3 ndash 8x2 + 13x ndash 10 by x ndash 2

remainder

Example (on your handout) Use synthetic division to divide 2x4 + 5x2 ndash 3 by x ndash 5

If powers of x are missing from the dividend there must be a placeholder in the synthetic division for each missing term

The Remainder amp Factor Theorem

The Remainder Theorem states that

If a polynomial f(x)1113089 is divided by x ndash k the remainder is r = f(1113089k)1113089

To evaluate a polynomial function when x = k divide the function by x ndash k (synthetic division)

Example 5Use the Remainder Theorem to evaluate f(x) = 4x3 + 10x2 ndash 3x ndash 8 at the value f(frac12)

Perform synthetic division

Example 5Use the Remainder Theorem to evaluate f(x) = 4x3 + 10x2 ndash 3x ndash 8 at the value f(frac12)

Check with substitution

Example (on your handout)

Use the Remainder Theorem to evaluate f(x) = 2x4 + 5x2 ndash 3 at the value x = 5

Perform synthetic division Check with substitution

The Factor Theorem states that a polynomial f(x) has a factor (x ndash k) iff f(k) = 0

To use the Factor Theorem to show that (x minus k) is a factor of a polynomial function f(x)

If the result is 0 then (x ndash k) is a factor

Example 6 Show that (x + 3) is a factor of f(x) = 3x3 + 7x2 ndash 3x + 9 Then find the remaining factor of f(x)

What have you learned

List three facts about the remainder r obtained in the synthetic division of f(x) by x minus k

• Polynomial amp Synthetic Division
• Important Vocabulary
• Long Division of Polynomials
• Example (on your handout)
• Synthetic Divison
• Example 4
• Example (on your handout) (2)
• The Remainder amp Factor Theorem
• Example 5
• Example 5 (2)
• Example (on your handout)
• Slide 12
• Example 6
• What have you learned

Important Vocabulary

Improper ndash the degree of the numerator is greater than or equal to the degree of the denominator

Proper ndash the degree of the numerator is less than the degree of the denominator

Long Division of Polynomials

Dividing polynomials is useful when ________________________

When dividing a polynomial f(x) by another polynomial d(x) if the remainder r(x) = 0 d(x) _____________________ into f(x)

The result of a division problem can be checked by _________________

evaluating functions

divides evenly

multiplication

Example (on your handout)

Divide 3x3 + 4x ndash 2 by x2 + 2x + 1

What can I multiply x2 by to get 3x3

Multiply 3x by x2 + 2x + 1

Subtract

What can I multiply x2 by to get -6x2

Multiply -6 by x2 + 2x + 1

Subtract

Synthetic Divison

Synthetic division is a shortcut for long division of polynomials by divisors of the form x ndash k

Example 4 Use synthetic division to divide 2x3 ndash 8x2 + 13x ndash 10 by x ndash 2

remainder

Example (on your handout) Use synthetic division to divide 2x4 + 5x2 ndash 3 by x ndash 5

If powers of x are missing from the dividend there must be a placeholder in the synthetic division for each missing term

The Remainder amp Factor Theorem

The Remainder Theorem states that

If a polynomial f(x)1113089 is divided by x ndash k the remainder is r = f(1113089k)1113089

To evaluate a polynomial function when x = k divide the function by x ndash k (synthetic division)

Example 5Use the Remainder Theorem to evaluate f(x) = 4x3 + 10x2 ndash 3x ndash 8 at the value f(frac12)

Perform synthetic division

Example 5Use the Remainder Theorem to evaluate f(x) = 4x3 + 10x2 ndash 3x ndash 8 at the value f(frac12)

Check with substitution

Example (on your handout)

Use the Remainder Theorem to evaluate f(x) = 2x4 + 5x2 ndash 3 at the value x = 5

Perform synthetic division Check with substitution

The Factor Theorem states that a polynomial f(x) has a factor (x ndash k) iff f(k) = 0

To use the Factor Theorem to show that (x minus k) is a factor of a polynomial function f(x)

If the result is 0 then (x ndash k) is a factor

Example 6 Show that (x + 3) is a factor of f(x) = 3x3 + 7x2 ndash 3x + 9 Then find the remaining factor of f(x)

What have you learned

List three facts about the remainder r obtained in the synthetic division of f(x) by x minus k

• Polynomial amp Synthetic Division
• Important Vocabulary
• Long Division of Polynomials
• Example (on your handout)
• Synthetic Divison
• Example 4
• Example (on your handout) (2)
• The Remainder amp Factor Theorem
• Example 5
• Example 5 (2)
• Example (on your handout)
• Slide 12
• Example 6
• What have you learned

Long Division of Polynomials

Dividing polynomials is useful when ________________________

When dividing a polynomial f(x) by another polynomial d(x) if the remainder r(x) = 0 d(x) _____________________ into f(x)

The result of a division problem can be checked by _________________

evaluating functions

divides evenly

multiplication

Example (on your handout)

Divide 3x3 + 4x ndash 2 by x2 + 2x + 1

What can I multiply x2 by to get 3x3

Multiply 3x by x2 + 2x + 1

Subtract

What can I multiply x2 by to get -6x2

Multiply -6 by x2 + 2x + 1

Subtract

Synthetic Divison

Synthetic division is a shortcut for long division of polynomials by divisors of the form x ndash k

Example 4 Use synthetic division to divide 2x3 ndash 8x2 + 13x ndash 10 by x ndash 2

remainder

Example (on your handout) Use synthetic division to divide 2x4 + 5x2 ndash 3 by x ndash 5

If powers of x are missing from the dividend there must be a placeholder in the synthetic division for each missing term

The Remainder amp Factor Theorem

The Remainder Theorem states that

If a polynomial f(x)1113089 is divided by x ndash k the remainder is r = f(1113089k)1113089

To evaluate a polynomial function when x = k divide the function by x ndash k (synthetic division)

Example 5Use the Remainder Theorem to evaluate f(x) = 4x3 + 10x2 ndash 3x ndash 8 at the value f(frac12)

Perform synthetic division

Example 5Use the Remainder Theorem to evaluate f(x) = 4x3 + 10x2 ndash 3x ndash 8 at the value f(frac12)

Check with substitution

Example (on your handout)

Use the Remainder Theorem to evaluate f(x) = 2x4 + 5x2 ndash 3 at the value x = 5

Perform synthetic division Check with substitution

The Factor Theorem states that a polynomial f(x) has a factor (x ndash k) iff f(k) = 0

To use the Factor Theorem to show that (x minus k) is a factor of a polynomial function f(x)

If the result is 0 then (x ndash k) is a factor

Example 6 Show that (x + 3) is a factor of f(x) = 3x3 + 7x2 ndash 3x + 9 Then find the remaining factor of f(x)

What have you learned

List three facts about the remainder r obtained in the synthetic division of f(x) by x minus k

• Polynomial amp Synthetic Division
• Important Vocabulary
• Long Division of Polynomials
• Example (on your handout)
• Synthetic Divison
• Example 4
• Example (on your handout) (2)
• The Remainder amp Factor Theorem
• Example 5
• Example 5 (2)
• Example (on your handout)
• Slide 12
• Example 6
• What have you learned

Example (on your handout)

Divide 3x3 + 4x ndash 2 by x2 + 2x + 1

What can I multiply x2 by to get 3x3

Multiply 3x by x2 + 2x + 1

Subtract

What can I multiply x2 by to get -6x2

Multiply -6 by x2 + 2x + 1

Subtract

Synthetic Divison

Synthetic division is a shortcut for long division of polynomials by divisors of the form x ndash k

Example 4 Use synthetic division to divide 2x3 ndash 8x2 + 13x ndash 10 by x ndash 2

remainder

Example (on your handout) Use synthetic division to divide 2x4 + 5x2 ndash 3 by x ndash 5

If powers of x are missing from the dividend there must be a placeholder in the synthetic division for each missing term

The Remainder amp Factor Theorem

The Remainder Theorem states that

If a polynomial f(x)1113089 is divided by x ndash k the remainder is r = f(1113089k)1113089

To evaluate a polynomial function when x = k divide the function by x ndash k (synthetic division)

Example 5Use the Remainder Theorem to evaluate f(x) = 4x3 + 10x2 ndash 3x ndash 8 at the value f(frac12)

Perform synthetic division

Example 5Use the Remainder Theorem to evaluate f(x) = 4x3 + 10x2 ndash 3x ndash 8 at the value f(frac12)

Check with substitution

Example (on your handout)

Use the Remainder Theorem to evaluate f(x) = 2x4 + 5x2 ndash 3 at the value x = 5

Perform synthetic division Check with substitution

The Factor Theorem states that a polynomial f(x) has a factor (x ndash k) iff f(k) = 0

To use the Factor Theorem to show that (x minus k) is a factor of a polynomial function f(x)

If the result is 0 then (x ndash k) is a factor

Example 6 Show that (x + 3) is a factor of f(x) = 3x3 + 7x2 ndash 3x + 9 Then find the remaining factor of f(x)

What have you learned

List three facts about the remainder r obtained in the synthetic division of f(x) by x minus k

• Polynomial amp Synthetic Division
• Important Vocabulary
• Long Division of Polynomials
• Example (on your handout)
• Synthetic Divison
• Example 4
• Example (on your handout) (2)
• The Remainder amp Factor Theorem
• Example 5
• Example 5 (2)
• Example (on your handout)
• Slide 12
• Example 6
• What have you learned

Synthetic Divison

Synthetic division is a shortcut for long division of polynomials by divisors of the form x ndash k

Example 4 Use synthetic division to divide 2x3 ndash 8x2 + 13x ndash 10 by x ndash 2

remainder

Example (on your handout) Use synthetic division to divide 2x4 + 5x2 ndash 3 by x ndash 5

If powers of x are missing from the dividend there must be a placeholder in the synthetic division for each missing term

The Remainder amp Factor Theorem

The Remainder Theorem states that

If a polynomial f(x)1113089 is divided by x ndash k the remainder is r = f(1113089k)1113089

To evaluate a polynomial function when x = k divide the function by x ndash k (synthetic division)

Example 5Use the Remainder Theorem to evaluate f(x) = 4x3 + 10x2 ndash 3x ndash 8 at the value f(frac12)

Perform synthetic division

Example 5Use the Remainder Theorem to evaluate f(x) = 4x3 + 10x2 ndash 3x ndash 8 at the value f(frac12)

Check with substitution

Example (on your handout)

Use the Remainder Theorem to evaluate f(x) = 2x4 + 5x2 ndash 3 at the value x = 5

Perform synthetic division Check with substitution

The Factor Theorem states that a polynomial f(x) has a factor (x ndash k) iff f(k) = 0

To use the Factor Theorem to show that (x minus k) is a factor of a polynomial function f(x)

If the result is 0 then (x ndash k) is a factor

Example 6 Show that (x + 3) is a factor of f(x) = 3x3 + 7x2 ndash 3x + 9 Then find the remaining factor of f(x)

What have you learned

List three facts about the remainder r obtained in the synthetic division of f(x) by x minus k

• Polynomial amp Synthetic Division
• Important Vocabulary
• Long Division of Polynomials
• Example (on your handout)
• Synthetic Divison
• Example 4
• Example (on your handout) (2)
• The Remainder amp Factor Theorem
• Example 5
• Example 5 (2)
• Example (on your handout)
• Slide 12
• Example 6
• What have you learned

Example 4 Use synthetic division to divide 2x3 ndash 8x2 + 13x ndash 10 by x ndash 2

remainder

Example (on your handout) Use synthetic division to divide 2x4 + 5x2 ndash 3 by x ndash 5

If powers of x are missing from the dividend there must be a placeholder in the synthetic division for each missing term

The Remainder amp Factor Theorem

The Remainder Theorem states that

If a polynomial f(x)1113089 is divided by x ndash k the remainder is r = f(1113089k)1113089

To evaluate a polynomial function when x = k divide the function by x ndash k (synthetic division)

Example 5Use the Remainder Theorem to evaluate f(x) = 4x3 + 10x2 ndash 3x ndash 8 at the value f(frac12)

Perform synthetic division

Example 5Use the Remainder Theorem to evaluate f(x) = 4x3 + 10x2 ndash 3x ndash 8 at the value f(frac12)

Check with substitution

Example (on your handout)

Use the Remainder Theorem to evaluate f(x) = 2x4 + 5x2 ndash 3 at the value x = 5

Perform synthetic division Check with substitution

The Factor Theorem states that a polynomial f(x) has a factor (x ndash k) iff f(k) = 0

To use the Factor Theorem to show that (x minus k) is a factor of a polynomial function f(x)

If the result is 0 then (x ndash k) is a factor

Example 6 Show that (x + 3) is a factor of f(x) = 3x3 + 7x2 ndash 3x + 9 Then find the remaining factor of f(x)

What have you learned

List three facts about the remainder r obtained in the synthetic division of f(x) by x minus k

• Polynomial amp Synthetic Division
• Important Vocabulary
• Long Division of Polynomials
• Example (on your handout)
• Synthetic Divison
• Example 4
• Example (on your handout) (2)
• The Remainder amp Factor Theorem
• Example 5
• Example 5 (2)
• Example (on your handout)
• Slide 12
• Example 6
• What have you learned

Example (on your handout) Use synthetic division to divide 2x4 + 5x2 ndash 3 by x ndash 5

If powers of x are missing from the dividend there must be a placeholder in the synthetic division for each missing term

The Remainder amp Factor Theorem

The Remainder Theorem states that

If a polynomial f(x)1113089 is divided by x ndash k the remainder is r = f(1113089k)1113089

To evaluate a polynomial function when x = k divide the function by x ndash k (synthetic division)

Example 5Use the Remainder Theorem to evaluate f(x) = 4x3 + 10x2 ndash 3x ndash 8 at the value f(frac12)

Perform synthetic division

Example 5Use the Remainder Theorem to evaluate f(x) = 4x3 + 10x2 ndash 3x ndash 8 at the value f(frac12)

Check with substitution

Example (on your handout)

Use the Remainder Theorem to evaluate f(x) = 2x4 + 5x2 ndash 3 at the value x = 5

Perform synthetic division Check with substitution

The Factor Theorem states that a polynomial f(x) has a factor (x ndash k) iff f(k) = 0

To use the Factor Theorem to show that (x minus k) is a factor of a polynomial function f(x)

If the result is 0 then (x ndash k) is a factor

Example 6 Show that (x + 3) is a factor of f(x) = 3x3 + 7x2 ndash 3x + 9 Then find the remaining factor of f(x)

What have you learned

List three facts about the remainder r obtained in the synthetic division of f(x) by x minus k

• Polynomial amp Synthetic Division
• Important Vocabulary
• Long Division of Polynomials
• Example (on your handout)
• Synthetic Divison
• Example 4
• Example (on your handout) (2)
• The Remainder amp Factor Theorem
• Example 5
• Example 5 (2)
• Example (on your handout)
• Slide 12
• Example 6
• What have you learned

The Remainder amp Factor Theorem

The Remainder Theorem states that

If a polynomial f(x)1113089 is divided by x ndash k the remainder is r = f(1113089k)1113089

To evaluate a polynomial function when x = k divide the function by x ndash k (synthetic division)

Example 5Use the Remainder Theorem to evaluate f(x) = 4x3 + 10x2 ndash 3x ndash 8 at the value f(frac12)

Perform synthetic division

Example 5Use the Remainder Theorem to evaluate f(x) = 4x3 + 10x2 ndash 3x ndash 8 at the value f(frac12)

Check with substitution

Example (on your handout)

Use the Remainder Theorem to evaluate f(x) = 2x4 + 5x2 ndash 3 at the value x = 5

Perform synthetic division Check with substitution

The Factor Theorem states that a polynomial f(x) has a factor (x ndash k) iff f(k) = 0

To use the Factor Theorem to show that (x minus k) is a factor of a polynomial function f(x)

If the result is 0 then (x ndash k) is a factor

Example 6 Show that (x + 3) is a factor of f(x) = 3x3 + 7x2 ndash 3x + 9 Then find the remaining factor of f(x)

What have you learned

List three facts about the remainder r obtained in the synthetic division of f(x) by x minus k

• Polynomial amp Synthetic Division
• Important Vocabulary
• Long Division of Polynomials
• Example (on your handout)
• Synthetic Divison
• Example 4
• Example (on your handout) (2)
• The Remainder amp Factor Theorem
• Example 5
• Example 5 (2)
• Example (on your handout)
• Slide 12
• Example 6
• What have you learned

Example 5Use the Remainder Theorem to evaluate f(x) = 4x3 + 10x2 ndash 3x ndash 8 at the value f(frac12)

Perform synthetic division

Example 5Use the Remainder Theorem to evaluate f(x) = 4x3 + 10x2 ndash 3x ndash 8 at the value f(frac12)

Check with substitution

Example (on your handout)

Use the Remainder Theorem to evaluate f(x) = 2x4 + 5x2 ndash 3 at the value x = 5

Perform synthetic division Check with substitution

The Factor Theorem states that a polynomial f(x) has a factor (x ndash k) iff f(k) = 0

To use the Factor Theorem to show that (x minus k) is a factor of a polynomial function f(x)

If the result is 0 then (x ndash k) is a factor

Example 6 Show that (x + 3) is a factor of f(x) = 3x3 + 7x2 ndash 3x + 9 Then find the remaining factor of f(x)

What have you learned

List three facts about the remainder r obtained in the synthetic division of f(x) by x minus k

• Polynomial amp Synthetic Division
• Important Vocabulary
• Long Division of Polynomials
• Example (on your handout)
• Synthetic Divison
• Example 4
• Example (on your handout) (2)
• The Remainder amp Factor Theorem
• Example 5
• Example 5 (2)
• Example (on your handout)
• Slide 12
• Example 6
• What have you learned

Example 5Use the Remainder Theorem to evaluate f(x) = 4x3 + 10x2 ndash 3x ndash 8 at the value f(frac12)

Check with substitution

Example (on your handout)

Use the Remainder Theorem to evaluate f(x) = 2x4 + 5x2 ndash 3 at the value x = 5

Perform synthetic division Check with substitution

The Factor Theorem states that a polynomial f(x) has a factor (x ndash k) iff f(k) = 0

To use the Factor Theorem to show that (x minus k) is a factor of a polynomial function f(x)

If the result is 0 then (x ndash k) is a factor

Example 6 Show that (x + 3) is a factor of f(x) = 3x3 + 7x2 ndash 3x + 9 Then find the remaining factor of f(x)

What have you learned

List three facts about the remainder r obtained in the synthetic division of f(x) by x minus k

• Polynomial amp Synthetic Division
• Important Vocabulary
• Long Division of Polynomials
• Example (on your handout)
• Synthetic Divison
• Example 4
• Example (on your handout) (2)
• The Remainder amp Factor Theorem
• Example 5
• Example 5 (2)
• Example (on your handout)
• Slide 12
• Example 6
• What have you learned

Example (on your handout)

Use the Remainder Theorem to evaluate f(x) = 2x4 + 5x2 ndash 3 at the value x = 5

Perform synthetic division Check with substitution

The Factor Theorem states that a polynomial f(x) has a factor (x ndash k) iff f(k) = 0

To use the Factor Theorem to show that (x minus k) is a factor of a polynomial function f(x)

If the result is 0 then (x ndash k) is a factor

Example 6 Show that (x + 3) is a factor of f(x) = 3x3 + 7x2 ndash 3x + 9 Then find the remaining factor of f(x)

What have you learned

List three facts about the remainder r obtained in the synthetic division of f(x) by x minus k

• Polynomial amp Synthetic Division
• Important Vocabulary
• Long Division of Polynomials
• Example (on your handout)
• Synthetic Divison
• Example 4
• Example (on your handout) (2)
• The Remainder amp Factor Theorem
• Example 5
• Example 5 (2)
• Example (on your handout)
• Slide 12
• Example 6
• What have you learned

The Factor Theorem states that a polynomial f(x) has a factor (x ndash k) iff f(k) = 0

To use the Factor Theorem to show that (x minus k) is a factor of a polynomial function f(x)

If the result is 0 then (x ndash k) is a factor

Example 6 Show that (x + 3) is a factor of f(x) = 3x3 + 7x2 ndash 3x + 9 Then find the remaining factor of f(x)

What have you learned

List three facts about the remainder r obtained in the synthetic division of f(x) by x minus k

• Polynomial amp Synthetic Division
• Important Vocabulary
• Long Division of Polynomials
• Example (on your handout)
• Synthetic Divison
• Example 4
• Example (on your handout) (2)
• The Remainder amp Factor Theorem
• Example 5
• Example 5 (2)
• Example (on your handout)
• Slide 12
• Example 6
• What have you learned

Example 6 Show that (x + 3) is a factor of f(x) = 3x3 + 7x2 ndash 3x + 9 Then find the remaining factor of f(x)

What have you learned

List three facts about the remainder r obtained in the synthetic division of f(x) by x minus k

• Polynomial amp Synthetic Division
• Important Vocabulary
• Long Division of Polynomials
• Example (on your handout)
• Synthetic Divison
• Example 4
• Example (on your handout) (2)
• The Remainder amp Factor Theorem
• Example 5
• Example 5 (2)
• Example (on your handout)
• Slide 12
• Example 6
• What have you learned

What have you learned

List three facts about the remainder r obtained in the synthetic division of f(x) by x minus k

• Polynomial amp Synthetic Division
• Important Vocabulary
• Long Division of Polynomials
• Example (on your handout)
• Synthetic Divison
• Example 4
• Example (on your handout) (2)
• The Remainder amp Factor Theorem
• Example 5
• Example 5 (2)
• Example (on your handout)
• Slide 12
• Example 6
• What have you learned