## Product Rule of Exponents

Multiplication of the same constant or variable to various powers can be simplified by adding up their exponents:

$5^3 \times 5^4 = 5^7$

This also holds true for distribution:

$x^2(x-2x^2) = x^3-2x^4$ because $x = x^1$

$4x^2 \times 9x^7 = 36x^9$

$x^3(2x+3x^2) = 2x^4 + 3x^5$ because $x^3 = 1 \times x^3$

$(x-y)(x-y)^3 = (x-y)^4$

## Power Rule of Exponents

Exponents distribute with multiplication:

$(5^3)^4 = 5^{12}$

$(b^m)^n = b^{m \times n}$

Signs are to be interpreted as $-1$:

$(-3x^4)^3 = (-1 \times 3 \times x^4)^3$

$= (-1)^3 3^3 x^{12}$

$= (-1) \times 27 \times x^{12}$

The product rule is not used when constants are not the same.

## Polynomials and Monomials

Polynomials are expressions linking multiple terms by addition and/or subtraction.

### Degree

The degree of a term is the sum of all of its exponents. A polynomial's degree is the highest degree of the degrees of its terms.

### Exceptions

An expression containing any of the following is *not* a polynomial:

- a fraction with a variable in the denominator
- any negative exponents

$(7x^2+3x-8)-(5x^2+6) = 7x^2+3x-8-5x^2-6$

$= 2x^2 + 3x - 14$

### Distribution of Monomials

$3y^3(-7y^4) = -21y^7$

$-y(-2y+4) = 2y - 4y$

$2xy(7x^2y^3-3y) = 14x^3y^4-6xy^2$

$a^4(3a-7) = 3a^5 - 7a^4$

### Distribution of Binomials

All terms are to be multiplied together with the resulting terms linked by either addition or subtraction:

$(x+3)(x+5) = (x \times x)+(x \times 5)+(3 \times x)+(3 \times 5)$

$= x^2 + 8x + 15$

### Special Products of Polynomials

- $(a-b)(a+b) = a^2-b^2$ -> $(2x^2-5y^3)(2x^2+5y^3) = 4x^4 - 25y^6$
- $(a+b)^2 = (a+b)(a+b) = a^2+ab+ab+b^2 = a^2+2ab+b^2$ -> $(2y+4)^2 = 4y^2+16y+16$

### Dividing Polynomials by Monomials

Considering that $\frac 1 7 + \frac 3 7 = \frac 4 7$, divisions of polynomials by *monomials* can be interpreted as the sum of two fractions with the *same denominator*:

$\frac {x^5 + x^3}{x^2} = \frac {x^5}{x^2} + \frac {x^3}{x^2} = x^3 + x$

### Long Division Algorithm for Polynomials

In some cases long division is the best way to handle the division of a polynomial by another polynomial:

$\frac {2x^3-x+1}{x+1} = x+1 \overline{)2x^3+0x^2-x+1} = 2x^2-2x+1$

- Fill up missing powers
- Divide the first term on the right side ($2x^3$) by the first term on the left side ($x$)
- Move result to the top of the line
- Distribute the same term with the terms on the left side, writing the result below the term on the right (while filling up missing powers)
- Subtract the polynomial below the line with the new one written beneath it
- Bring down the next term if it isn't already part of the result
- Repeat
- If the algorithm's last subtraction does not yield a $0$, it has yielded a
*remainder*. The remainder is brought up as numerator over the left polynomial (in this case $x+1$ as its denominator.

## Negative Exponents

Negative exponents can be transformed into positive numbers by turning their base into the numerator of a fraction and turning that fraction into its *reciprocal*:

$b^{-n}$ -> $\frac {b^{-n}}{1} = \frac {1}{b^n}$ -> $9^{-2} = \frac {1}{9^2} = \frac {1}{81}$

$(x-y)^{-7} = \frac {1}{(x-y)^7}$

$3x^{-1} = \frac {3}{x^1}$

$5xy^{-1} = \frac {5x}{y^1}$

## Quotient Rule for Exponents

A number raised to a power divided by the same number is equal to that itself raised to the power of the exponents' difference.

$\frac {b^m}{b^n} = b^{m-n}$ -> $\frac {x^3}{x^2} = x$

$\frac {x^5}{x^7} = x^{-2} = \frac {1}{x^2}$
$\frac {28x^6y^2}{7x^3y^7} = 4x^3 \frac {1}{y^5}$

$\frac {1}{b^{-n}} = b^n$ -> $\frac {1}{5^{-4}} = 5^4$

$\frac {a^{-m}}{b^{-n}} = \frac {b^n}{a^m}$

$(\frac a b)^{-n} = \frac {a^{-n}}{b^{-n}} = \frac {b^n}{a^n}$

### Complex terms with fractions

These can be interpreted as multiple individual fractions multiplied together:

$\frac {7x^{-3}y^{-5}}{x^{-3}y^{-2}} = \frac 7 1 \times \frac {x^{-3}}{x^{-3}} \times \frac {y^{-5}}{y^{-2}} = \frac 7 1 \times 1 \times \frac {1}{y^3}$