XIII Algebraic Exponent Rules and Polynomials

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

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

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

1. Fill up missing powers
2. Divide the first term on the right side ($2x^3$) by the first term on the left side ($x$)
3. Move result to the top of the line
4. 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)
5. Subtract the polynomial below the line with the new one written beneath it
6. Bring down the next term if it isn't already part of the result
7. Repeat
8. 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:
$\frac {7x^{-3}y^{-5}}{x^{-3}y^{-2}} = \frac 7 1 \times \frac {x^{-3}}{x^{-3}} \times \frac {y^{-5}}{y^{-2}} = \frac 7 1 \times 1 \times \frac {1}{y^3}$