Algebra 2 Honors: Polynomial Theorems, Remainder & Factor Theorems, and Binomial Expansion

Relationship Between Zeros, Degree, and Intercepts:
- When all zeros of a function are real numbers, the graph has a number of $x$ -intercepts equal to the number of real zeros (for example, if all zeros are real, a polynomial with $3$ real zeros will have $3$ $x$ -intercepts).
- When some or all of the zeros of a function are complex numbers, the number of $x$ -intercepts will not be equal to the degree of the function.
- The degree of the function will always be equal to the total number of zeros (counting both real and complex zeros).
Fundamental Definitions:
- Theorem: A statement, not necessarily mathematical in nature, proven by experimentation.
- The Fundamental Theorem of Algebra: Generally states that the degree of a polynomial is equivalent to the number of zeros (both real and complex) of a function.
Detailed Example:
- Function: $f(x) = x^2 + 11x + 18$
- Degree of Function: $2$
- Total Number of Zeros: $2$
- Real Zeros: $x = -9$ and $x = -2$
- Core Principle: Real zeros are identical to the $x$ -intercepts of the function's graph.
Graph Analysis Question:
- Examine the graph of the function $f(x) = x^3 - 12x^2 + 35x$ . What are the zeros?
Module & Course Resources:
- Module Topic: H1.01 Theorems - Note taking and videos
- Extra Help/Videos URL: https://sites.google.com/flvs.net/algebra-2-videosandhelp/segment-1-honors-module/honors-1 -01-theorems

Definition:
- The Factor Theorem: States that a first degree binomial is a factor of a polynomial function if the remainder, when the polynomial is divided by the binomial, is zero.
Determining Factors (Worked Procedure):
- Problem: Determine whether $x - 5$ is a factor of the function $f(x) = -4x^3 + 21x^2 - 25$ .
- Setup: Set up a division problem where $f(x) = -4x^3 + 21x^2 - 25$ is divided by $x - 5$ .
- Remainder Evaluation: When $f(x) = -4x^3 + 21x^2 - 25$ is divided by the binomial $x - 5$ , the remainder is $0$ .
- Conclusion: Because the remainder is zero, $x - 5$ is a factor of the function.
Example 1:
- Problem Prompt: Is $x + 12$ a factor of the function $f(x) = x^2 + 6x - 72$ ? Explain.
- Instruction: Go through the slides and take notes below.

Definition:
- The Remainder Theorem: States that when the opposite of the constant from the binomial divisor is substituted into a function for $x$ , the result is the remainder.
Comparing Division and Substitution Methods:
- Using Division: When the polynomial function $f(x) = x^4 + 11x^3 + 26x^2 + 15x - 17$ is divided by $x + 8$ using division, the remainder is the last integer on the bottom row.
- Using Substitution: When the opposite of the constant in the divisor is substituted into the function, the result will be identical to the remainder obtained through the division process.
- Solving for the Substitution Constant:
- Divisor equation: $x + 8 = 0$
- Subtract $8$ from both sides: $x + 8 - 8 = 0 - 8$
- Opposite constant: $x = -8$
Example 1:
- Problem Prompt: Find the remainder when $f(x) = 5x^2 + 51x + 16$ is divided by $x + 10$ .
- Method 1: Using Division
- Method 2: Using Substitution

Overview & Definition:
- To expand a binomial $(a + b)^n$ , where $n$ is a whole number, the coefficients follow Pascal's Triangle.
Exponent Rules During Expansion:
- With each successive term in the expansion, the power of $a$ decreases by $1$ .
- With each successive term in the expansion, the power of $b$ increases by $1$ .
Pascal's Triangle Binomial Expansions:
- Degree $0$ : $(a + b)^0 = 1$
- Degree $1$ : $(a + b)^1 = 1a + 1b$
- Degree $2$ : $(a + b)^2 = 1a^2 + 2ab + 1b^2$
- Degree $3$ : $(a + b)^3 = 1a^3 + 3a^2b + 3ab^2 + 1b^3$
- Degree $4$ : $(a + b)^4 = 1a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + 1b^4$
Expansion Practice Problems:
- Example 1: Expand $(x + 2)^5$ using the Binomial Theorem and Pascal's triangle.
- Example 2: Expand $(x - 3)^4$ using the Binomial Theorem and Pascal's triangle.
Additional Notes:
- Reserved for extra lecture notes and practice exercises.