1. Synthetic Division (Fast Polynomial Division)
When to use it:
- When dividing a polynomial by a linear factor of the form $(x - a)$ or $(x + a)$
- When checking whether a value is a root
- When finding missing factors
How it works:
- Write the root (opposite sign of the factor) on the left.
- Write all coefficients, including zeros for missing terms.
- Bring the first coefficient straight down.
- Multiply the bottom value by the root.
- Add vertically.
- Continue until the end.
Example: factor $(x + 4)$ → root is $-4$
Remember: Use the opposite sign of what appears in the factor!
A remainder of 0 means the factor is real.
2. Factoring Polynomials Using Synthetic Division
Steps:
- Use the known factor to perform synthetic division.
- The bottom row (excluding remainder) becomes the coefficient list of the quotient (one degree lower).
- Factor the resulting quadratic (or repeat synthetic division).
- Write the full factorization as a product of linear factors.
If synthetic division gives:
$$[1, 3, -2, 0]$$Then the quotient is:
$$x^2 + 3x - 2$$Which may further factor into $(x + 4)(x - 1)$ or similar.
3. Using Roots to Find Linear Factors
A polynomial's linear factors come from its zeros:
- If $x = a$ is a root → factor is $(x - a)$
- If a factor is given → the root is its opposite sign
Quick evaluation test:
To check whether $(x - a)$ is a factor, plug $a$ into the polynomial:
- If $P(a) = 0$ → it is a factor
- If $P(a) \neq 0$ → it is not a factor
💡 Pro Tip: This trick is crucial for multiple-choice factor identification.
4. Finding Constant Terms from Factored Form
To find the constant term of a polynomial written in factored form:
- Plug in $x = 0$
- Multiply the constants from each factor
- Apply any numerical coefficient outside the factors
Example:
$$P(x) = -5(x - 2)(x - 3)(x + 7)$$Constant term:
$$-5(0 - 2)(0 - 3)(0 + 7) = -5(-2)(-3)(7)$$5. Solving Equations in Factored Form
When an equation is of the form $(\text{expression}) = \text{number}$, move everything to one side so it equals 0.
Example: $(x - 2)(x - 4) = 8$
Step 1: Bring 8 over
$$(x - 2)(x - 4) - 8 = 0$$Step 2: Expand
$$x^2 - 6x + 8 - 8 = 0$$Step 3: Factor
$$x(x - 6) = 0$$Step 4: Solve each factor
$$x = 0, \quad x = 6$$6. End Behavior of Polynomial Functions
End behavior depends ONLY on:
- The degree (odd or even)
- The leading coefficient (positive or negative)
| Degree | Leading Coefficient | Left End | Right End |
|---|---|---|---|
| Odd | Positive | Down ↓ | Up ↑ |
| Odd | Negative | Up ↑ | Down ↓ |
| Even | Positive | Up ↑ | Up ↑ |
| Even | Negative | Down ↓ | Down ↓ |
📊 Example from a graph: If the graph falls left and rises right → odd degree, positive leading coefficient.
7. Graphing a Polynomial from Factored Form
When the polynomial is factored:
You immediately know:
- Roots: $x = r_1, r_2, r_3,\dots$
- X-intercepts: these same points
- End behavior: from the leading term (highest degree) and the sign of $a$
- General shape: smooth curve, no sharp corners, number of turns ≤ degree – 1
Example to graph: $(x + 4)(x + 1)(x - 2)$
Intercepts:
$$x = -4, \quad x = -1, \quad x = 2$$Behavior:
- Left → down ↓
- Right → up ↑
Sketch accordingly.
8. Common Mistakes to Avoid
- ❌ Forgetting a zero coefficient when a term is missing
- ❌ Incorrect sign when adding during synthetic division
- ❌ Assuming a factor is correct without verifying with the root
- ❌ Mixing up $(x - a)$ and $(x + a)$
✅ Remember: Always double-check your work by plugging roots back into the original polynomial!