Polynomial & Synthetic Division Algebra III, Sec. 2.3 Objective Use long division and synthetic...
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Transcript of Polynomial & Synthetic Division Algebra III, Sec. 2.3 Objective Use long division and synthetic...
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
-