Alex Notes
Chapter 1 Fundamental Algebraic Ideas
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closed (closure) Addition is unique and \[ a+b\in\R \]
Multiplication is unique and \[ ab\in\R \]
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commutative (commutation) addition \[ a+b=b+a \]
multiplication \[ ab=ba \]
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associative (association) addition \[ (a+b)+c=a+(b+c) \] multiplication \[ (ab)c=a(bc) \]
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identity addition \[ a+0=0+a=a \] multiplication \[ a(1)=(1)a=a \]
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inverse addition \[ a+(-a)=(-a)+a=0 \] multiplication \[ a\cdot\frac{1}{a} =\frac{1}{a}\cdot a =1\quad(a\ne0) \]
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distributive (distribution) left distributive \[ a(b+c)=ab+ac \] right distributive \[ (a+b)c=ac+bc \]
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multiplicative property of $0$ \[ a(0)=(0)a=0 \]
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multiplicative property of $-1$ \[ a(-1)=(-1)a=-a \]
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negative of a sum \[ -(a+b)=-a-b \]
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negative of a difference \[ -(a-b)=b-a \]
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distributive property over subtraction \[ a(b-c)=ab-ac\quad\text{(left)}\\ (a-b)c=ac-bc\quad\text{(right)} \]
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negative in products \[ (-a)b=-(ab)=a(-b)\\ (-a)(-b)=ab \]
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negative in quotients \[ \frac{-a}{b}=-\frac{a}{b}=\frac{a}{-b}\\ \frac{-a}{-b}=\frac{a}{b} \]
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addition property for all $\frac{a}{b},$ $\frac{c}{b},$ $\frac{c}{d},$ $b\ne0,$ $d\ne0$\[ \frac{a}{b}+\frac{c}{b}=\frac{a+c}{b} \]
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subtraction property \[ \frac{a}{b}-\frac{c}{b}=\frac{a-c}{b} \]
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multiplication property \[ \frac{a}{b}\cdot\frac{c}{d}=\frac{ac}{bd} \]
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division property where $c\ne0$\[ \frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\cdot\frac{d}{c} \]
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product property \[ a^ma^n=a^{m+n} \]
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power property \[ (a^m)^n=a^{mn} \]
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quotient property \[ \frac{a^m}{a^n}= \left\{ \begin{array}{cr} a^{m-n}&\text{if }m\gt n\\ \frac{1}{a^{n-m}}&\text{if }m\lt n\\ 1&\text{if }m=n \end{array} \right. \]
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power of a product property \[ (ab)^n=a^nb^n \]
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power of a quotient property \[ \left(\frac{a}{b}\right)^n=\frac{a^n}{b^n} \]
- \[ a^0=1 \]
- \[ a^{-n}=\frac{1}{a^n} \]
- \[ a^m a^n = a^{m+n} \]
- \[ (a^m)^n=a^{mn} \]
- \[ \frac{a^m}{a^n}=a^{m-n} \]
- \[ (ab)^n=a^nb^n \]
- \[ \left(\frac{a}{b}^n\right)=\frac{a^n}{b^n} \]
- \[ \frac{a^{-m}}{b^{-n}}=\frac{b^n}{a^m} \]
| $n$ | $a$ | $n\text{th}$ root(s) of $a$ |
|---|---|---|
| even | positive | two real roots: one positive, one negative |
| even | negative | no real root |
| odd | positive | one real root: a positive root |
| odd | negative | one real root: a negative root |
| odd or even | zero | one real root: $0$ |
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If $a$ is positive and $n$ is even, then
- $\sqrt[n]{a}$ is the
positive or principal $n\text{th}$ root of $a$ -
$-\sqrt[n]{a}$ is
the negative of the principal $n\text{th}$ root of $a.$ - $\sqrt[n]{-a}$ is not a real number.
- $\sqrt[n]{a}$ is the
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If $a$ is either positive or negative and $n$ is odd,
then $\sqrt[n]{a}$ represents
the $n\text{th}$ root of $a.$ - $\sqrt[n]{0}=0$ for all positive integers $n.$
- \[ (\sqrt[n]{a})^n=a \]
- \[ \left(\sqrt[m]{\sqrt[n]{a}}\right)=\sqrt[mn]{a} \]
- \[ \sqrt[n]{ab}=\sqrt[n]{a}\sqrt[n]{b} \]
- \[ \sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}} \quad(b\ne0) \]
- The radicand contains no factor to a power greater than or equal to the index of the radical; that is, $\sqrt[n]{a^m}$ has $m\lt n.$
- The power of the radicand and the index of the radical have no common factor other than $1;$ that is, for $\sqrt[n]{a^m},$ the power $m$ and the index $n$ are relatively prime.
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square of a binomial, perfect square trinomial \[ (A+B)^2=\\ A^2+2AB+B^2 \] -
cube of a binomial, perfect cube polynomial \[ (A+B)^3=\\ A^3+3A^2B+3AB^2+B^3 \] -
product of conjugate pairs, difference of squares \[ (A+B)(A-B)\\ =A^2-B^2 \] -
sum of cubes \[ (A+B)(A^2-AB+B^2)\\ =A^3+B^3 \] -
difference of cubes \[ (A-B)(A^2+AB+B^2)\\ =A^3-B^3 \]
The
- Integers
- Rational Numbers
- Irrational Numbers
- Real Numbers
- Pure Imaginary Numbers
- Imaginary Numbers
Note these relationships:
- Pure Imaginary Numbers $\subset$ Imaginary Numbers
- Imaginary Numbers $\subset$ Complex Numbers
- Integers $\subset$ Rational Numbers $\subset$ Real Numbers
- Rational Numbers $\subset$ Real Numbers
- Irrational Numbers $\subset$ Real Numbers
- Real Numbers $\subset$ Complex Numbers
- Rational Numbers $\cap$ Irrational Numbers = $\varnothing.$
- Imaginary Numbers $\cap$ Real Numbers = $\varnothing.$
- Use the distributive property.
- Group terms.
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Factor Trinomials
- Case 1. If $k_1$ and $k_2$ are constants, then the product of the binomials $(x+k_1)$ and $(x+k_2)$ is a trinomial of the form $x^2+bx+c,$ where $b=k_1+k_2$ and $c=k_1k_2.$ Thus, to factor trinomials of the form $x^2+bx+c,$ we find two numbers $k_1$ and $k_2$ whose product is $c$ and whose sum is $b.$ We then write \[ x^2+bx+c=\\ (x+k_1)(x+k_2) \]
- Case 2. To factor trinomials of the form $ax^2+bx+c,$ $(a\ne1),$ we find two numbers whose product is $ac$ and whose sum is $b.$ If these two numbers are $k_1$ and $k_2,$ then $ac=k_1k_2$, $bx=k_1x + k_2x$ and we write \[ ax^2+bx+c=\\ ax^2+k_1x+k_2x+c, \] then factor this expression by grouping terms.
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Case 3.
perfect square trinomial. In the trinomial $Ax^2+Bx+C,$ if the middle term is twice the product of the square root of the first and last terms, then $A=a^2,$ $B=ac$, and $C=c^2$ for some $a,b,c\in\R$. Thus, by the previous technique for factoring a trinomial we obtain the factorization directly by writing \[ Ax^2+Bx+C=\\ (ax)^2+(2ab)x+c^2 =\\(ax+b)^2 \]
- Trial and error.
To add or subtract algebraic fractions
with unlike denominators, we first find
the
- Completely factor each denominator, and use exponential notation to represent repeated prime factors that occur in any one of the denominators.
- The product of each different prime factor to the highest power it occurs in any one of the denominators is the LCD.