📈 Advanced Polynomial Operations

Example: $(x^3 + 2x^2 - 5x + 2) \div (x - 1)$

Step-by-step process:

1. Divide the leading term: $x^3 \div x = x^2$
2. Multiply: $x^2(x - 1) = x^3 - x^2$
3. Subtract: $(x^3 + 2x^2) - (x^3 - x^2) = 3x^2$
4. Bring down next term: $3x^2 - 5x$
5. Repeat until done!

$$\text{Result: } x^2 + 3x - 2$$

Example: $(2x^3 - 3x^2 + 5x - 1) \div (x - 2)$

Setup: Use 2 (from $x - 2$) and coefficients [2, -3, 5, -1]

2 |  2  -3   5  -1
  |     4   2  14
  |-----------------
     2   1   7  13

Result: $2x^2 + x + 7$ with remainder $13$

So: $2x^3 - 3x^2 + 5x - 1 = (x - 2)(2x^2 + x + 7) + 13$

Example

Find the remainder when $P(x) = x^3 - 2x^2 + 5x - 7$ is divided by $(x - 3)$

Using the Remainder Theorem:

Just evaluate $P(3)$!

$P(3) = 3^3 - 2(3)^2 + 5(3) - 7$

$= 27 - 18 + 15 - 7$

$$= 17$$

The remainder is 17! ✨

Example: Is $(x - 2)$ a factor of $P(x) = x^3 - 6x^2 + 11x - 6$?

Test: Evaluate $P(2)$

$P(2) = 2^3 - 6(2)^2 + 11(2) - 6$

$= 8 - 24 + 22 - 6 = 0$ ✅

Yes! Since $P(2) = 0$, $(x - 2)$ IS a factor!

Finding Other Factors

Use synthetic division to find: $P(x) = (x - 2)(x^2 - 4x + 3)$

Factor further: $= (x - 2)(x - 1)(x - 3)$

The Theorem

For $P(x) = a_nx^n + ... + a_1x + a_0$, any rational root $\frac{p}{q}$ must have:

• $p$ divides $a_0$ (constant term)
• $q$ divides $a_n$ (leading coefficient)

Example: $2x^3 - 3x^2 - 11x + 6 = 0$

Possible values of $p$: $\pm1, \pm2, \pm3, \pm6$ (factors of 6)

Possible values of $q$: $\pm1, \pm2$ (factors of 2)

Possible rational roots: $\pm1, \pm2, \pm3, \pm6, \pm\frac{1}{2}, \pm\frac{3}{2}$