1. Apply the Remainder and Factor Theorems Lesson 2.5
Honors Algebra 2
Apply the Remainder and Factor Theorems
Lesson 2.5

2. Goals Goal Rubric Use polynomial long division. The Remainder Theorem.
Use synthetic division.
The Factor Theorem.
Factor a polynomial using the factor theorem and synthetic division.
Level 1 – Know the goals.
Level 2 – Fully understand the goals.
Level 3 – Use the goals to solve simple problems.
Level 4 – Use the goals to solve more advanced problems.
Level 5 – Adapts and applies the goals to different and more complex problems.

3. Vocabulary
Polynomial long division
Synthetic division

4. Polynomial Long Division
Polynomial long division is a method for dividing a polynomial by another polynomials of a lower degree. It is very similar to dividing numbers.

5. Polynomial Long Division
Procedure:
Arrange the terms of both polynomials in descending order of the exponents of the variable.
If either the dividend or the divisor has missing terms, insert these terms with coefficients of 0.
Divide
Check, by multiplying the divisor times the quotient and adding the remainder.

6. Example: Divide 4x + 2x3 – 1 by 2x – 2
+ 3
Write the terms of the dividend in descending order.
2x3 – 2x2
2x2
+ 4x
Since there is no x2 term in the dividend, add 0x2 as a placeholder.
2x2 – 2x 6x
– 1 1. 2.
6x – 6 4. 3. 5
Answer: x2 + x + 3 5 5. 6. 7. 8. 9.

7. Divide f (x) = 3x4 – 5x3 + 4x – 6 by x2 – 3x + 5.
EXAMPLE 1
Use polynomial long division
Divide f (x) = 3x4 – 5x3 + 4x – 6 by x2 – 3x + 5.
SOLUTION
Write polynomial division in the same format you use when dividing numbers. Include a “0” as the coefficient of x2 in the dividend. At each stage, divide the term with the highest power in what is left of the dividend by the first term of the divisor. This gives the next term of the quotient.

8. ) 3x2 + 4x – 3 x2 – 3x + 5 3x4 – 5x3 + 0x2 + 4x – 6 3x4 – 9x3 + 15x2
EXAMPLE 1
Use polynomial long division
3x2
+ 4x
– 3
quotient
x2 – 3x + 5
3x4 – 5x3 + 0x2 + 4x – 6 )
Multiply divisor by 3x4/x2 = 3x2
3x4 – 9x3 + 15x2
4x3 – 15x2 + 4x
Subtract. Bring down next term.
Multiply divisor by 4x3/x2 = 4x
4x3 – 12x2 + 20x
– 3x2 – 16x – 6
Subtract. Bring down next term.
Multiply divisor by – 3x2/x2 = – 3
–3x2 + 9x – 15
– 25x + 9
remainder

9. 3x4 – 5x3 + 4x – 6 x2 – 3x + 5 = 3x2 + 4x – 3 + – 25x + 9 EXAMPLE 1
Use polynomial long division
3x4 – 5x3 + 4x – 6
x2 – 3x + 5
= 3x2 + 4x – 3 +
– 25x + 9
ANSWER

10. ) Divide f (x) = x3 + 5x2 – 7x + 2 by x – 2. + 7x + 7 x – 2
EXAMPLE 2
Use polynomial long division with a linear divisor
Divide f (x) = x3 + 5x2 – 7x + 2 by x – 2. x2
+ 7x
+ 7
quotient
x – 2
x3 + 5x2 – 7x )
x3 – 2x2
Multiply divisor by x3/x = x2.
7x2 – 7x
Subtract.
Multiply divisor by 7x2/x = 7x.
7x2 – 14x
7x + 2
Subtract.
Multiply divisor by 7x/x = 7.
7x – 14 16
remainder
ANSWER
x3 + 5x2 – 7x +2
x – 2
= x2 + 7x + 7 + 16

11. 1. (2x4 + x3 + x – 1) (x2 + 2x – 1) 2x4 + x3 + x – 1 x2 + 2x – 1
Your Turn:
for Examples 1 and 2
Divide using polynomial long division.
(2x4 + x3 + x – 1) (x2 + 2x – 1)
2x4 + x3 + x – 1
x2 + 2x – 1
= (2x2 – 3x + 8) +
– 18x + 7
ANSWER
2. (x3 – x2 + 4x – 10) ÷ (x + 2)
x3 – x2 +4x – 10
x + 2
= (x2 – 3x +10) +
– 30
ANSWER

12. Synthetic Division
Synthetic division is a shorthand method of dividing a polynomial by a linear binomial by using only the coefficients. For synthetic division to work, the polynomial must be written in standard form, using 0 and a coefficient for any missing terms, and the divisor must be in the form (x – a).

13. Synthetic Division

14. Remainder Theorem What it means: Used to find the remainder.
When dividing a polynomial by a divisor of the form (x-k), the Remainder Theorem gives us two ways of calculating the remainder.
Use synthetic division to divide by (x-k).
Evaluate the function at f(k).

15. Synthetic Division
Use synthetic division to find the quotient:
(x3 – 3x2 + x + 6) ÷ (x – 3)
Step 1: Write the coefficients of the polynomial, and the k-value of the divisor on the left 3 1 -3 1 6

16. Synthetic Division
Use synthetic division to find the quotient:
(x3 – 3x2 + x + 6) ÷ (x – 3)
Step 2: Draw a line and write the first coefficient under the line. 3 1 -3 1 6 1

17. Synthetic Division
Use synthetic division to find the quotient:
(x3 – 3x2 + x + 6) ÷ (x – 3)
Step 3: Multiply the k-value, 3, by the number below the line and write the product below the next coefficient. 3 1 -3 1 6 3 1

18. Synthetic Division
Use synthetic division to find the quotient:
(x3 – 3x2 + x + 6) ÷ (x – 3)
Step 4: Write the sum of -3 and 3 below the line. 3 1 -3 1 6 3 1

19. Synthetic Division
Use synthetic division to find the quotient:
(x3 – 3x2 + x + 6) ÷ (x – 3)
Repeat steps 3 and 4. 3 1 -3 1 6 3 1 1

20. Synthetic Division
Use synthetic division to find the quotient:
(x3 – 3x2 + x + 6) ÷ (x – 3)
Repeat steps 3 and 4. 3 1 -3 1 6 3 3 1 1 9

21. Synthetic Division 3 1 -3 1 6 3 3 1 1 9 x2 + 1 + x – 3 9
Use synthetic division to find the quotient:
(x3 – 3x2 + x + 6) ÷ (x – 3)
The remainder is 9 and the resulting numbers are the coefficients of the quotient. 3 1 -3 1 6 3 3 1 1 9 x
x – 3 9

22. Divide f (x)= 2x3 + x2 – 8x + 5 by x + 3 using synthetic division.
EXAMPLE 3
Use synthetic division
Divide f (x)= 2x3 + x2 – 8x + 5 by x + 3 using synthetic
division.
SOLUTION
– –
– – 21
2 – – 16
2x3 + x2 – 8x + 5
x + 3
= 2x2 – 5x + 7 – 16
ANSWER

23. 3. (x3 + 4x2 – x – 1) ÷ (x + 3) x3 + 4 x2 – x – 1 x + 3 = x2 + x – 4 +
Your Turn:
for Examples 3 and 4
Divide using synthetic division.
3. (x3 + 4x2 – x – 1) ÷ (x + 3)
x3 + 4 x2 – x – 1
x + 3
= x2 + x – 4 + 11
ANSWER
4. (4x3 + x2 – 3x + 7) ÷ (x – 1)
4x3 + x2 – 3x + 7
x – 1
= 4x2 + 5x + 2 + 9
ANSWER

24. Factor Theorem
Recall that if a number is divided by any of its factors, the remainder is 0. Likewise, if a polynomial is divided by any of its factors, the remainder is 0.
The Remainder Theorem states that if a polynomial is divided by (x – a), the remainder is the value of the function at a. So, if (x – a) is a factor of P(x), then P(a) = 0.

25. Factor f (x) = 3x3 – 4x2 – 28x – 16 completely given that
EXAMPLE 4
Factor a polynomial
Factor f (x) = 3x3 – 4x2 – 28x – 16 completely given that
x + 2 is a factor.
SOLUTION
Because x + 2 is a factor of f (x), you know that f (– 2) = 0. Use synthetic division to find the other factors.
– – 4 – – 16 –
3 – –

26. Use the result to write f (x) as a product of two
EXAMPLE 4
Factor a polynomial
Use the result to write f (x) as a product of two
factors and then factor completely.
f (x) = 3x3 – 4x2 – 28x – 16
Write original polynomial.
= (x + 2)(3x2 – 10x – 8)
Write as a product of two
factors.
= (x + 2)(3x + 2)(x – 4)
Factor trinomial.

27. Factor the polynomial completely given that x – 4 is a factor.
Your Turn:
for Examples 3 and 4
Factor the polynomial completely given that x – 4 is a factor.
f (x) = x3 – 6x2 + 5x + 12
ANSWER
f (x) = (x – 4)(x –3)(x + 1)
f (x) = x3 – x2 – 22x + 40
ANSWER
f (x) = (x – 4)(x –2)(x +5)

28. Because f (3) = 0, x – 3 is a factor of f (x). Use synthetic division.
EXAMPLE 5
Standardized Test Practice
SOLUTION
Because f (3) = 0, x – 3 is a factor of f (x). Use synthetic division.
– 2 –
– 60 –

29. Use the result to write f (x) as a product of two
EXAMPLE 5
Standardized Test Practice
Use the result to write f (x) as a product of two
factors. Then factor completely.
f (x) = x3 – 2x2 – 23x + 60
= (x – 3)(x2 + x – 20)
= (x – 3)(x + 5)(x – 4)
The zeros are 3, – 5, and 4.
The correct answer is A.
ANSWER

30. The profit P (in millions of dollars) for a shoe manufacturer can be
EXAMPLE 6
Use a polynomial model
BUSINESS
The profit P (in millions
of dollars) for a shoe
manufacturer can be
modeled by
P = – 21x3 + 46x
where x is the number
of shoes produced
(in millions). The company
now produces 1 million shoes and makes a profit of $25,000,000, but would like to cut back production. What lesser number of shoes could the company produce and still make the same profit?

31. EXAMPLE 6
Use a polynomial model
SOLUTION
25 = – 21x3 + 46x
Substitute 25 for P in P = – 21x3 + 46x.
0 = 21x3 – 46x + 25
Write in standard form.
You know that x = 1 is one solution of the equation. This implies that x – 1 is a factor of 21x3 – 46x Use synthetic division to find the other factors. –
–25 –

32. The company could still make the same profit
EXAMPLE 6
Use a polynomial model
So, (x – 1)(21x2 + 21x – 25) = 0. Use the quadratic formula to find that x ≈ 0.7 is the other positive solution.
The company could still make the same profit
producing about 700,000 shoes.
ANSWER

33. Find the other zeros of f given that f (– 2) = 0.
Your Turn:
for Examples 5 and 6
Find the other zeros of f given that f (– 2) = 0.
f (x) = x3 + 2x2 – 9x – 18
ANSWER
The other zeros are 3 and – 3.
f (x) = x3 + 8x2 + 5x – 14
ANSWER
The other zeros are 1 and – 7.

34. The company could still make the same profit
Your Turn:
for Examples 5 and 6 9.
What if? In Example 6, how does the answer change if the profit for the shoe manufacturer is modeled by P = – 15x3 + 40x?
The company could still make the same profit
producing about 900,000 shoes.
ANSWER