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# Synthetic Division

See how to do the synthetic division
to factor a polynomial
and solve a polynomial equation/inequality.
13 examples and their solutions.

## Synthetic Division

### Example

x3 - 7x + 11x - 2
Solution

### Example

2x4 + x3 - 5x2 + 3x + 4x + 1
Solution

## Remainder Theorem

### Formula

The remainder of f(x)x - a
→ f(a)
By using the remainder theorem,
you can find the remainder of f(x)/(x - a)
without doing the whole division.
Just find f(a).
(a: Zero of x - a)

### Example

x3 - 7x + 11x - 2
Remainder?
Solution

### Example

(2x4 + x3 - 5x2 + 3x + 4) ÷ (x + 1)
Remainder?
Solution

## Synthetic Substitution

### Formula

f(x)
a...
f(a)
When doing the synthetic division,
the remainder is the right bottom number.
And by the remainder theorem,
the remainder of f(x)/(x - a) is f(a).
So (right bottom number) = (remainder) = f(a).
So you can find f(a)
by finding the remainder of the synthetic division.
When f(x) is complex,
this method will save your time.

### Example

f(x) = x4 - 9x3 + 15x2 + 3x - 62
f(7) = ?
Solution

## Factor Theorem

### Theorem

If f(a) = 0,
then f(x) = (x - a)(quotient).
If f(a) = 0,
then, by the remainder theorem,
the remainder of f(x)/(x - a) is 0.
So f(x) = (x - a)(quotient).
This is the factor theorem.
If f(a) = 0, f(b) = 0, ... ,
then f(x) = (x - a)(x - b)(quotient).
So, if a and b are the zeros of f(x),
then f(x) = (x - a)(x - b)(quotient).

### Formula

f(x)
a...
0
b...
(quotient)0

f(x) = (x - a)(x - b)(quotient)
Let's mix the factor theorem
and the synthetic division.
To factor f(x),
do the synthetic division
and find the zeros [a, b, ...]
that makes the remainder 0.

### Example

x3 + 3x2 - 16x + 12
Solution

### Example

x4 - 2x3 - 4x2 - 2x + 3
Solution

### Example

x4 + x3 - 5x2 + x - 6
Solution

## Polynomial Equation

### Example

x4 + 4x3 - 3x2 - 10x + 8 = 0
Solution

## Polynomial Inequality

### Formula

(x - a)odd(x - a)even
If f(x) = (x - a)odd(quotient),
then y = f(x) passes through the x-axis
at x = a.
If f(x) = (x - a)even(quotient),
then y = f(x) bounces off the x-axis
at x = a.
Use this formula
when graphing y = f(x) on the x-axis
to solve a polynomial inequality.

x4 - x2 < 0
Solution

### Example

x3 + x2 - 10x + 8 ≥ 0
Solution

### Example

x4 + x3 - 5x2 + 3x ≤ 0
Solution

x4 - x < 0
Solution