Download Math III Unit 2 Day 7 Synthetic Division

Guided Notes: Synthetic Division, Remainder Theorem, &
Fundamental Theorem of Algebra
Synthetic Division
Step 1: Write the degrees in DESCENDING order.
Step 2: Bring down the coefficients.
Step 3: Write the constant of the divisor to the left box.
Step 4: Bring the FIRST coefficient down.
Step 5: Multiply the constant & coefficient and write under the NEXT coefficient.
Step 6: Add the column.
Step 7: Repeat this process starting at Step 5
Example1: (𝑥 3 + 2𝑥 2 − 5𝑥 + 16) ÷ (𝑥 − 4)
Example 2:(𝑥 3 + 27) ÷ (𝑥 + 3)
Example 3: (2𝑛3 + 62𝑛 − 26𝑛2 + 4) ÷ (2𝑛 − 6)
Discovery Activity: Remainder and Factor Theorem
1.a If 𝑓(𝑥) = (2𝑥 2 − 13𝑥 + 15), divide f(x) by (𝑥 − 5). What is the remainder?
b. Now calculate 𝑓(5).
2.a. If 𝑓(𝑥) = (4𝑥 2 − 11𝑥 − 3), divide f(x) by (𝑥 + 6). What is the remainder?
b. Now calculate 𝑓(−6).
Summarize: What do you notice? Do you see a pattern?
Remainder Theorem-
What does it mean to be a factor? A polynomial is a factor of a polynomial when the
remainder is _____________. (Look at 1.a.)
How can we use the remainder theorem to determine if a polynomial is a factor??
Factor Theorem-
3. Let 𝑝(𝑥) = 𝑥 3 − 𝑥 4 + 8𝑥² − 9𝑥 + 30. Evaluate 𝑝(– 2). What does the solution tell
you about the factors of 𝑝(𝑥)?
4. Consider the polynomial function: 𝑃(𝑥) = 𝑥 4 − 3𝑥 3 + 𝑎𝑥² − 6𝑥 + 14, where a is an
unknown real number. If (𝑥 − 2) is a factor of this polynomial, what is the value of a?
Fundamental Theorem of Algebra
If P(x) is a polynomial of degree n, then P(x) = 0 has exactly n roots, including multiple and
complex roots.
What does this mean?
Remember: Irrational and imaginary zeros come in ___________________.
Thus a linear equation has ______ solution, a quadratic has ______ solutions, a cubic has ______
solutions, and so on. The zeroes do not have to be unique. For instance (𝑥 − 3)² = 0 has
zeroes at 𝑥 = 3 and 𝑥 = 3. This is considered to have a double root or a multiplicity of two.
So far, we have been using graphing to find the REAL zeros. What if we want to find all
EXACT REAL and IMAGINARY zeros??
Rational Zero Theorem-P/Q Method
List all the possible rational zeros of each function.
a. 𝑓(𝑥) = 𝑥 3 + 3𝑥 2 − 𝑥 + 8
b. ℎ(𝑥) = 𝑥 4 − 7𝑥 3 − 4𝑥 2 + 𝑥 − 49
c. 𝑓(𝑥) = 3𝑥 4 − 2𝑥 2 + 6𝑥 − 10
Steps to finding all the zeros of a polynomial: Real and Imaginary!
1. Find all possible rational zeros (using p/q method)
2. Type equation into y= on your calculator
3. Click 2nd Window, Indpnt: Ask
4. 2nd Graph
5. Type in possible zeros, see which value gives you zero
6. Use synthetic division to keep breaking down polynomial
Example 1: Find all zeros of 𝑓(𝑥) = 𝑥 3 + 6𝑥 2 + 4𝑥 + 24
Example 2: Find all zeros of 𝑓(𝑥) = 4𝑥 4 + 5𝑥 3 + 30𝑥 2 + 45𝑥 − 54
Example 3: Find all zeros of 𝑓(𝑥) = 2𝑥 4 − 2𝑥 3 − 45𝑥 2 − 36𝑥 + 81
Homework
Divide the following polynomials using synthetic division.
1. (𝑥 3 − 20) ÷ (𝑥 − 3)
2. (10𝑟 3 + 50𝑟 2 − 60) ÷ (10𝑟 − 10)
Homework
1. What is the remainder when we divide 𝑓(𝑥) = 2𝑥 2 + 4𝑥 − 5 by (𝑥 + 2)?
2. What is the remainder when we divide 𝑓(𝑥) = 3𝑥 3 + 4𝑥 2 − 5𝑥 + 7 by (𝑥 − 4)?
3. What is the remainder when we divide 𝑔(𝑥) = 4𝑥 3 − 3𝑥 + 1 by (𝑥 − 1)?
3. Determine whether each number is a root of 𝑎4 − 13𝑎2 + 12𝑎 = 0. Then, explain how
you know.
a) 0
b) -1
c)1
d) -4
e) -3
4. Determine whether x – 1 is a factor of f (x) = 2x4 + 3x2 – 5x + 7.
5. Determine whether x + 4 is a factor of f (x) = 5x4 + 16x3 – 15x2 + 8x + 16.
6. Determine the value of k that would make (𝑥 − 1) a factor of (3𝑥 3 + 𝑥 2 + 2𝑘𝑥 + 4)
7. Suppose that 𝑓(𝑥) = 𝑥 6 − 3𝑥 2 + 12𝑥 − 𝑘. The remainder of the division of f(x) by
(𝑥 − 1) is −4. What is the remainder of the division of f(x) by (x+1)?
Find all the zeros of the following functions
8. 𝑓(𝑥) = 𝑥 3 + 4𝑥 2 + 7𝑥 + 28
9. 𝑓(𝑥) = 2𝑥 3 + 𝑥 2 + 1