Finding Zeros and Analyzing Factored Polynomials

A polynomial is said to be in factored form (or intercept form) when it is expressed as the product of its linear factors and a constant leading coefficient.
The general mathematical representation of a factored polynomial of degree $n$ is: $P(x) = a(x - r_1)(x - r_2)(x - r_3) \dots (x - r_n)$
- $\text{In this expression: } a \text{ represents the leading coefficient.}$
- $\text{The values } r_1, r_2, \dots, r_n \text{ represent the individual roots or zeros of the polynomial.}$
This form is highly valuable because it explicitly displays the x-intercepts (zeros) of the function without requiring additional factoring or use of the quadratic formula.

The fundamental logic used to find the zeros of a factored polynomial is the Zero Product Property.
Definition: If the product of two or more algebraic expressions is equal to zero ( $A \times B = 0$ ), then at least one of the individual expressions must be equal to zero ( $A = 0$ or $B = 0$ ).
Applying this to a polynomial $P(x) = (x - 2)(x + 5)$ , if we seek the values where $P(x) = 0$ , we solve each factor independently:
- $x - 2 = 0 \implies x = 2$
- $x + 5 = 0 \implies x = -5$
Even if a leading coefficient is present, such as in $f(x) = -3(x - 1)(x + 4)$ , the constant ( $-3$ ) does not affect the location of the zeros because $-3 \neq 0$ . Only the factors containing the variable $x$ yield roots.

Step 1: Set the Polynomial Expression to Zero
- If given a function such as $y = (x + 8)(x - 11)$ , start by setting the entire equation to zero: $0 = (x + 8)(x - 11)$ .
Step 2: Identify Individual Factors
- Locate every expression inside parentheses that contains the variable $x$ .
Step 3: Solve for x in Each Factor
- For a factor in the form $(x - c)$ , the zero is $x = c$ .
- For a factor in the form $(x + c)$ , the zero is $x = -c$ .
- Mathematical Proof: $x + c = 0$ $x = 0 - c$ $x = -c$
Step 4: List the Zeros
- On platforms like IXL (My IXL Learning), zeros are typically entered as a list separated by commas.
- Example: If the factors result in $x = 4$ and $x = -9$ , the final answer is $4, -9$ .

The Concept of Multiplicity:
- Multiplicity refers to how many times a particular root is repeated in the factorization.
- If a factor is raised to a power, such as $(x - 3)^2$ , the root $x = 3$ has a multiplicity of 2.
- Graphical Behavior:
  - If the multiplicity is odd, the graph of the polynomial will cross the x-axis at that point.
  - If the multiplicity is even, the graph will "touch" or "bounce off" the x-axis at that point without crossing through it.
Determining the Degree of the Polynomial:
- To find the degree of a polynomial from its factored form, sum the exponents of all linear factors containing $x$ .
- For example, in $P(x) = x(x - 5)^3(x + 1)$ , the degree is calculated as: $1 (\text{for the } x) + 3 (\text{for the } (x - 5)^3) + 1 (\text{for the } (x + 1)) = 5$
- The degree tells you the maximum number of real zeros the polynomial can have.

Example A (Simple Linear Factors):
- Function: $f(x) = (x - 12)(x + 15)$
- Factors: $(x - 12) = 0 \implies x = 12$ , $(x + 15) = 0 \implies x = -15$
- Zeros: $12, -15$
Example B (With Fractions and Coefficients):
- Function: $g(x) = (2x - 1)(x + 3)$
- Solving for the first factor: $2x - 1 = 0 \implies 2x = 1 \implies x = \frac{1}{2}$
- Solving for the second factor: $x + 3 = 0 \implies x = -3$
- Zeros: $\frac{1}{2}, -3$
Example C (Variable as an Isolated Factor):
- Function: $h(x) = -x(x - 4)$
- The leading factor is $x$ . Setting it to zero gives $x = 0$ .
- The second factor is $(x - 4) = 0 \implies x = 4$ .
- Zeros: $0, 4$