1. Chapter 7.4 The Remainder and Factor Theorems Standard & Honors
Algebra II Mr. Gilbert
Chapter 7.4
The Remainder and Factor Theorems
Standard & Honors
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2. Agenda
Warm up
Home Work
Lesson
Practice
Homework
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3. Homework Review
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4. Communicate Effectively
Synthetic substitution: When synthetic division is used to evaluate a function.
Depressed polynomial: The quotient when a polynomial is divided by one of its binomial factors.
For example: divide
x4+x3-17x2-20x+32 by x-4
the depressed polynomial is x3+5x2+3x-8
The Factor Theorem: The binomial x-a is a factor of f(x) iff f(a)=0.
The Remainder Theorem: If a polynomial f(x) is divided by x-a, the remainder is the constant f(a) where q(x) is a polynomial with degree one less than f(x).
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5. Example 1 Synthetic Substitution (3)
Example 2 Use the Factor Theorem (3)
Example 3 Find All Factors of a Polynomial (3)
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Lesson 4 Contents

6. 3 10 41 164 654 If find f (4). Method 1 Synthetic Substitution
By the Remainder Theorem, f (4) should be the remainder when you divide the polynomial by x – 4.
Notice that there is no x term. A zero is placed in this position as a placeholder.
Answer: The remainder is 654. Thus, by using synthetic substitution, f (4) = 654.
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Example 4-1a

7. Method 2 Direct Substitution
Replace x with 4.
Original function
Replace x with 4.
Simplify.
or 654
Answer: By using direct substitution, f (4) = 654.
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Example 4-1b

8. If find f (3).
Answer: 34
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Example 4-1c

9. Show that is a factor of Then find the remaining factors of the polynomial.
The binomial x – 3 is a factor of the polynomial if 3 is a zero of the related polynomial function. Use the factor theorem and synthetic division.
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Example 4-2a

10. Since the remainder is 0, (x – 3) is a factor of the polynomial
Since the remainder is 0, (x – 3) is a factor of the polynomial. The polynomial can be factored as The polynomial is the depressed polynomial. Check to see if this polynomial can be factored.
Factor the trinomial.
Answer: So,
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Example 4-2b

11. Check. You can see that the graph of the related function
Check You can see that the graph of the related function crosses the x-axis at 3, –6, and –1. Thus,
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Example 4-2c

12. Show that is a factor of Then find the remaining factors of the polynomial.
Answer:
So, Since
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Example 4-2d

13. Geometry The volume of a rectangular prism is given by Find the missing measures.
The volume of a rectangular prism is You know that one measure is x – 2, so x – 2 is a factor of V(x).
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Example 4-3a

14. The quotient is . Use this to factor V(x).
Volume function
Factor.
Factor the trinomial
Answer: The missing measures of the prism are x + 4 and x + 5.
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Example 4-3b

15. Answer: The missing measures of the prism are x – 2 and x + 5.
Geometry The volume of a rectangular prism is given by Find the missing measures.
Answer: The missing measures of the prism are x – 2 and x + 5.
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Example 4-3c

16. Homework
See Syllabus 7.4
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