Algebra Review

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ALGEBRA HINTS

Order of Operations (PEMDAS):

PLEASE EXCUSE MY DEAR AUNT SALLY
Parentheses Exponents Mult. Div. Add. Subt.
    Left to right, perform either multiplication OR division Left to right, perform either addition OR subtraction

 

Example:   \(\displaystyle 20-16\div {{2}^{3}}\times \left( {3+2} \right)=20-\frac{{16}}{8}\times 5=20-\left( {2\times 5} \right)=10\)

 

Properties of Real Numbers:

Associative (groups): \(a+\left( {b+c} \right)=\left( {a+b} \right)+c,\,\,\,\,a\left( {bc} \right)=\left( {ab} \right)c\)

Commutative (ordering): \(a+b=b+a,\,\,\,\,ab=ba\)

Distributive (“pushing through”): \(a\left( {b+c} \right)=ab+ac,\,\,\,\,\,a\left( {b-c} \right)=ab-ac\)

Inverse: \(\displaystyle a+-a=0,\,\,\,\,\,a\times \frac{1}{a}=1\)

Identity: \(a+0=a,\,\,\,\,\,a\times 1=a\)

Zero Property: \(a\times 0=0\)

Scientific Notation:

Examples:  \(\displaystyle \begin{array}{c}7.9\times {{10}^{5}}=79,000\\7.9\times {{10}^{{-4}}}=.00079\end{array}\)

Count to the right (left) that many decimal places when the exponent is positive (negative).

Note that the first number must be \(\ge 1\) and \(<10\).

Operations:

\(\begin{align}a\left( {x+y} \right)=ax+ay\\\frac{{a+b}}{c}=\frac{a}{c}+\frac{b}{c}\\\frac{a}{b}+\frac{c}{d}=\frac{{ad+bc}}{{bd}}\\\frac{a}{b}\times \frac{c}{d}=\frac{{ac}}{{bd}}\\\frac{{\frac{a}{c}}}{{\frac{b}{d}}}=\frac{a}{c}\div \frac{b}{d}=\frac{a}{c}\times \frac{d}{b}=\frac{{ad}}{{cb}}\end{align}\)

Factorial: \(6!=6\times 5\times 4\times 3\times 2\times 1=720;\,\,\,\,\,1!=1;\,\,\,\,\,\,0!=1\)

Absolute Value: \(\left| 5 \right|=5;\,\,\,\,\,\,\left| {-5} \right|=5\)

Exponents and Radicals:

\(\displaystyle \begin{array}{c}{{(xy)}^{m}}={{x}^{m}}\cdot {{y}^{m}}\\{{\left( {\frac{x}{y}} \right)}^{m}}=\frac{{{{x}^{m}}}}{{{{y}^{m}}}}\\{{x}^{m}}\cdot {{x}^{n}}={{x}^{{m+n}}}\\\frac{{{{x}^{m}}}}{{{{x}^{n}}}}={{x}^{{m-n}}}\\{{\left( {{{x}^{m}}} \right)}^{n}}={{x}^{{mn}}}\\{{x}^{1}}=x;\,\,\,\,{{x}^{0}}=1\\\frac{1}{{{{x}^{m}}}}={{x}^{{-m}}}\\{{\left( {\frac{x}{y}} \right)}^{{-m}}}=\,{{\left( {\frac{y}{x}} \right)}^{m}}\end{array}\)    \(\displaystyle \begin{array}{c}\sqrt[n]{x}={{x}^{{\frac{1}{n}}}}\,\,(\text{for reals},\text{if }n\text{ is even, }\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x\text{ must be }0\text{ or positive)}\\\sqrt[n]{{xy}}=\sqrt[n]{x}\cdot \sqrt[n]{y}\\{{\left( {\sqrt[n]{x}} \right)}^{m}}=\,\sqrt[n]{{{{x}^{m}}}}={{x}^{{\frac{m}{n}}}}\\\sqrt[n]{{{{x}^{n}}}}=\,\left| x \right|\\\frac{x}{{\sqrt{y}}}=\frac{x}{{\sqrt{y}}}\cdot \frac{{\sqrt{y}}}{{\sqrt{y}}}=\frac{{x\sqrt{y}}}{y}\,\,(\text{Rationalize})\\\text{Use Conjugate}:\\\frac{x}{{x+\sqrt{y}}}=\frac{x}{{x+\sqrt{y}}}\cdot \frac{{x-\sqrt{y}}}{{x-\sqrt{y}}}=\frac{{x\left( {x-\sqrt{y}} \right)}}{{{{x}^{2}}-y}}\end{array}\)

Lines:

Slope of line going through points \(\left( {{{x}_{1}},{{y}_{1}}} \right)\) and \(\left( {{{x}_{2}},{{y}_{2}}} \right)\) is \(m=\frac{{{{y}_{2}}-{{y}_{1}}}}{{{{x}_{2}}-{{x}_{1}}}}\). Point-slope equation of line through point  with slope \(m\) is \(y-{{y}_{1}}=m\left( {x-{{x}_{1}}} \right)\). Slope-intercept equation of line with slope \(m\) and \(y\)-intercept \(b\) is \(y=mx+b\).

 

Examples: Find the equation that passes through points \(\left( {4,0} \right)\) and \(\left( {-2,12} \right)\). First find slope: \(m=\frac{{{{y}_{2}}-{{y}_{1}}}}{{{{x}_{2}}-{{x}_{1}}}}=\frac{{12-0}}{{-2-4}}=-2\). For point-slope equation, use either point: \(y-0=-2\left( {x-4} \right);\,\,\,y=-2\left( {x-4} \right)=-2x+8\).

 

For slope-intercept equation, find \(y\)-intercept (\(m\)), using either point: \(y=-2x+b;\,\,12=-2\left( {-2} \right)+b;\,\,\,b=8:\,\,\,y=-2x+8\).

 

Example of slope of 0: line \(y=5\) (horizontal line). Example of undefined slope: \(x=-3\) (vertical line).

 

Parallel lines have the same slope. Perpendicular lines have slopes that are negative reciprocals of each other, for example \(4\) and \(\displaystyle -\frac{1}{4}\).

Solving Linear Equations:

1. Combine like terms.

2. Get all variables on left, constants on right.

3. Isolate the variable by dividing what’s in front of it.

Example: \(\displaystyle \begin{align}5x-2x-5&=16\\3x-5&=16\\3x-5+5&=16+5\end{align}\)      \(\begin{array}{l}3x=21\\\frac{{3x}}{3}=\frac{{21}}{3}\\x=7\end{array}\)

For Inequalities, remember to change the sign when multiplying or dividing by a negative number!

For Systems of Equations, use either substitution (substitute the expression of one variable into the other equation) or linear elimination to first solve for one variable, then the other. Example of elimination:

\(\displaystyle \begin{array}{l}2x+5y=-1\,\,\,\,\,\text{multiply by}-3\\7x+3y=11\text{ }\,\,\,\,\,\text{multiply by }5\\-6x-15y=3\,\\\,\underline{{35x+15y=55}}\text{ }\\\,29x\,\,\,\,\,\,\,\,\,\,\,\,\,\,=58\\\,\,\,\,\,\,\,\,\,\,\,x=2\end{array}\)

Use first equation to get \(y\):  \(\displaystyle y=\frac{{-1-2\left( 2 \right)}}{5}=-1\). Solution is \(\left( {2,-1} \right)\).

Direct/Indirect Variation:

Direct Variation formula: \(\displaystyle y=kx\) or \(\displaystyle  \frac{{{{y}_{1}}}}{{{{x}_{1}}}}=\frac{{{{y}_{2}}}}{{{{x}_{2}}}}\)

 

Indirect or Inverse Variation formula: \(\displaystyle y=\frac{k}{x}\), or \(k=xy\)

\(k\) is a constant

 

Examples: \(y\) varies directly with \(x\). \(y=10\) when \(x=2\). What is \(y\) when \(x=4\)?

\(y=kx;\,\,10=k\left( 2 \right);\,\,k=5.\,\,\,\,\,y=5x;\,\,\,y=5\left( 4 \right)=20\)

(Or, \(\displaystyle \frac{{10}}{2}=\frac{{{{y}_{2}}}}{4};\,\,2{{y}_{2}}=40;\,\,{{y}_{2}}=20\))

 

\(y\) varies indirectly with \(x\). \(y=10\) when \(x=4\). What is \(y\) when \(x=2\)?

\(\displaystyle y=\frac{k}{x};\,\,10=\frac{k}{4};\,\,k=40.\,\,\,\,\,y=\frac{{40}}{x};\,\,\,y=\frac{{40}}{2}=20\)

Multiplying/Factoring Polynomials:

Multiplying Binomials:

\(\begin{array}{c}\left( {a+b} \right)\left( {c+d} \right)=ac+ad+bc+bd\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,F\,\,\,\,\,\,\,O\,\,\,\,\,\,\,I\,\,\,\,\,\,\,L\end{array}\)

Perfect Square Binomials:

\(\begin{array}{c}{{\left( {a+b} \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}\\{{\left( {a-b} \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}\end{array}\)

Difference of Squares:

\(\displaystyle \left( {a+b} \right)\left( {a-b} \right)={{a}^{2}}-{{b}^{2}}\)

Sum/Difference of Cubes:

\(\displaystyle \begin{array}{c}{{\left( {a+b} \right)}^{3}}=\left( {a+b} \right)\left( {{{a}^{2}}-ab+{{b}^{2}}} \right)\\{{\left( {a-b} \right)}^{3}}=\left( {a-b} \right)\left( {{{a}^{2}}+ab+{{b}^{2}}} \right)\end{array}\)

 

Quadratic Formula:

Quadratic formula gives roots of \(a{{x}^{2}}+bx+c=0\):

\(\displaystyle \frac{{-b\pm \sqrt{{{{b}^{2}}-4ac}}}}{{2a}}\)

 

Example:  For \({{x}^{2}}-4x-12\), \(a=1,\,\,b=-4,\,\,c=-12\).

Roots are:

\(\displaystyle \begin{align}x&=\frac{{-\left( {-4} \right)\pm \sqrt{{{{{\left( {-4} \right)}}^{2}}-4\left( 1 \right)\left( {-12} \right)}}}}{{2\left( 1 \right)}}=\frac{{4\pm \sqrt{{64}}}}{2}=\frac{{4\pm 8}}{2}\\x&=6,\,-2\end{align}\)

 

To get vertex, use \(\displaystyle \left( {-\frac{b}{{2a}},\,f\left( {-\frac{b}{{2a}}} \right)} \right)\) (plug in what you get for \(x\) to get \(y\)). \(\displaystyle x=-\frac{b}{{2a}}\) is the axis of symmetry. In above example, axis of symmetry is \(\displaystyle x=-\frac{b}{{2a}}=-\frac{{-4}}{{2\left( 1 \right)}}=2;\,\,\,\,x=2\). The vertex is \(\displaystyle \left( {2,{{{\left( 2 \right)}}^{2}}-4\left( 2 \right)-12} \right)=\left( {2,-16} \right)\).

Domain Restrictions (Undefined):

Domains are restricted to the following:

  • Denominator can’t be \(0\): \(\displaystyle \frac{a}{{\ne 0}}\)
  • Even root must be \(\ge 0\):  \(\sqrt[{\text{even}\,}]{{\,\ge 0}}\)
  • Argument of log must be \(>0\): \(\log \left( {>0} \right)\) or \(\ln \left( {>0} \right)\)

Examples: The domain of \(f\left( x \right)=\sqrt{{x+3}}\) is \(x+3\ge 0\), or \(x\ge -3\). The domain of \(f\left( x \right)=\log \left( {3x-2} \right)\) is \(3x-2>0\)or \(x>\frac{2}{3}\)The domain of \(f\left( x \right)=\frac{3}{{\sqrt{x}}}\) is \(x\ge 0\,\,\,\text{and}\,\,\,x\ne 0\), or \(x>0\).

Note that for a relation (equation) to be a function, it must pass the vertical line test, meaning that you can’t have more than one \(y\)-value for any \(x\)-value. For example, vertical lines aren’t functions, and neither are the points \(\left( {2,-5} \right)\,\,\text{and}\,\,\left( {2,0} \right)\).

 

To shift functions, make vertical shifts up and down based on what is added/subtracted on the outside, vertical stretches based on what is multiplied on the outside, and horizontal shifts in opposite direction based on what is added/subtracted on the inside.

Degrees of Polynomials:

To get the degree of a polynomial, add up all the exponents of each term (when multiplied out), and the degree is the highest number.

 

For example, for the polynomial \(2{{x}^{3}}{{y}^{2}}+x{{y}^{2}}-xy+6\), the degree is 5, since the first term’s sum of exponents is 5, the second is 3, the third is 2, and the last is 0.

 

For the polynomial \(x\left( {2x-3} \right)\left( {{{x}^{2}}+1} \right)\), the degree is 4, since the term with the highest exponent would be \(x\cdot 2x\cdot {{x}^{2}}=2{{x}^{4}}\).

Factoring and Roots:

FOIL backwards! Example: To factor \({{x}^{2}}-4x-12\), find two numbers that multiply to \(-12\) but add to \(-4\): \(-6\) and \(2\). Check, and it works!

\(\begin{array}{l}\,\,\,\,\,{{x}^{2}}-4x-12\\=\left( {x-6} \right)\left( {x+2} \right)\end{array}\)

 

To get roots, set each factor to \(0\):   \(\begin{array}{l}x-6=0;\,\,x=6\\x+2=0;\,\,x=-2\end{array}\)

 

To factor a quadratic like \(3{{x}^{2}}+10x-8\), either use “guess and check” or “ac” method. For the “ac” method, find two numbers that multiply to –24 (3 times –8), but add to 10 (coefficient of middle term): 12 and –2. Separate and factor:

\(\begin{array}{c}3{{x}^{2}}+10x-8\\3{{x}^{2}}\,\,\,\,\,+12x-2x\,\,\,\,\,\,-8\\3x\left( {x+4} \right)-2\left( {x+4} \right)\\\left( {3x-2} \right)\left( {x+4} \right)\end{array}\)

Permutations/Combinations:

Permutations:

Number of \(n\) things taken \(r\) at a time, order matters:

\(\displaystyle {}_{n}{{P}_{r}}=\frac{{n!}}{{\left( {n-r} \right)!}}\)

(Example: the number of ways to select a President and Vice-President out of 10 people is \({}_{{10}}{{P}_{2}}\).)

 

Combinations:

Number of \(n\) things taken \(r\) at a time, order doesn’t matter:

\(\displaystyle {}_{n}{{C}_{r}}=\frac{{n!}}{{r!\left( {n-r} \right)!}}\)

(Example: the number of ways to select two officers out of 10 people is \({}_{{10}}{{C}_{2}}\).)

Exponential Growth/Decay:

Exponential Growth: \(y=a{{b}^{x}},\,\,a>0,\,\,b>1\)

Exponential Decay: \(y=a{{b}^{x}},\,\,a>0,\,\,0<b<1\)

Compounding Formulas: \(A\) ending amount, \(P\) beginning amount, \(r\) annual rate, \(n\) number of times compounded per year, \(t\) number of years:

Growth: \(\displaystyle A=P{{\left( {1+\frac{r}{n}} \right)}^{{nt}}}\)     Decay: \(\displaystyle A=P{{\left( {1-\frac{r}{n}} \right)}^{{nt}}}\)

Simple Interest (linear, not exponential) growth: \(\displaystyle A=P\left( {1+rt} \right)\), Interest only is \(I=Prt\).

Continuous compounding (\(e\) is Euler’s number ): \(A=P{{e}^{{rt}}}\)

Half Life: \(y=a{{\left( {.5} \right)}^{{\frac{t}{p}}}}\), \(a\) beginning amount, \(y\) ending amount, \(t\) time, \(p\) half-life (time it takes to halve)

Logarithms:

\(y={{\log }_{b}}x\,\,\,\,\text{means}\,\,\,\,x={{b}^{y}}\,\,\,(b>0,\,\,b\ne 1,\,\,x>0)\)

 

Log Rules:

\(\begin{array}{c}{{\log }_{b}}\left( {xy} \right)={{\log }_{b}}x+{{\log }_{b}}y\\{{\log }_{b}}\left( {\frac{x}{y}} \right)={{\log }_{b}}x-{{\log }_{b}}y\\{{\log }_{b}}\left( {{{x}^{p}}} \right)=p{{\log }_{b}}x\end{array}\)       \(\begin{array}{c}\ln e\,\,\left( {={{{\log }}_{e}}\left( e \right)} \right)=1\\\ln {{e}^{x}}\,\,=x\\\ln \left( 1 \right)\,\,=0\\{{e}^{{\ln x}}}=x\end{array}\)

 

Take the ln or log of each side and then use the Power Rule when solving for variables in exponents. For example,

\(\displaystyle \begin{align}150{{\left( {.5} \right)}^{t}}&=1500\\{{\left( {.5} \right)}^{t}}&=\frac{{1500}}{{150}}=10\end{align}\)          \(\displaystyle \begin{align}\ln {{\left( {.5} \right)}^{t}}&=\ln \left( {10} \right)\\t\ln \left( {.5} \right)&=\ln \left( {10} \right)\\t&=\frac{{\ln \left( {10} \right)}}{{\ln \left( {.5} \right)}}\approx -3.32\end{align}\)

Distance/Midpoint Formulas:

Midpoint between points \(\left( {{{x}_{1}},{{y}_{1}}} \right)\) and \(\left( {{{x}_{2}},{{y}_{2}}} \right)\) is:

\(\displaystyle \left( {\frac{{{{x}_{1}}+{{x}_{2}}}}{2},\frac{{{{y}_{1}}+{{y}_{2}}}}{2}} \right)\)

Distance between points \(\left( {{{x}_{1}},{{y}_{1}}} \right)\) and \(\left( {{{x}_{2}},{{y}_{2}}} \right)\) is:

\(d=\sqrt{{{{{\left( {{{x}_{2}}-{{x}_{1}}} \right)}}^{2}}+{{{\left( {{{y}_{2}}-{{y}_{1}}} \right)}}^{2}}}}\)

 

Other formulas:

Pythagorean Theorem:   \({{a}^{2}}+{{b}^{2}}={{c}^{2}}\)

Equation of Circle with center \((h,k)\) and radius \(r\):  \({{\left( {x-h} \right)}^{2}}+{{\left( {y-k} \right)}^{2}}={{r}^{2}}\)

Formulas for Area, Circumference, Surface Area and Volume:

Triangle: \(\displaystyle A=\frac{1}{2}bh\)    Circle:  \(A=\pi {{r}^{2}};\,\,C=2\pi r\)

Sphere: \(\displaystyle SA=4\pi {{r}^{2}};\,\,\,V=\frac{4}{3}\pi {{r}^{2}}\)

Cylinder: \(\displaystyle SA=2\pi {{r}^{2}}+2\pi rh;\,\,\,V=\pi {{r}^{2}}h\)

Cone:  \(\displaystyle V=\frac{1}{3}\pi {{r}^{2}}h\)

Complex/Imaginary Numbers:

\({{i}^{2}}=-1;\,\,\,\,\,\,i=\sqrt{{-1}}\)

 

Example operations:

\(\displaystyle \begin{array}{c}6+i-\left( {4-2i} \right)=2+3i\\\left( {3+2i} \right)\left( {3-2i} \right)=9-6i+6i-{{\left( {2i} \right)}^{2}}=9-4{{i}^{2}}=9+4=13\end{array}\)

\(\displaystyle \frac{4}{{1+i}}=\,\frac{4}{{1+i}}\cdot \frac{{1-i}}{{1-i}}=\frac{{4-4i}}{{1-{{i}^{2}}}}=\frac{{4-4i}}{{1+1}}=2-2i\,\,\,\,\,\,\text{(Rationalize)}\)

 

\(\begin{align}\sqrt{{-3}}\cdot \sqrt{{-24}}&=\sqrt{3}i\cdot \sqrt{{24}}i\\&=\sqrt{3}i\cdot 2\sqrt{6}i\\&=2\sqrt{{18}}\cdot {{i}^{2}}\\&=-2\cdot 3\sqrt{2}\\&=-6\sqrt{2}\end{align}\)

Sequences and Series:

\(n=\) term number, \({{a}_{n}}=nth\) term, \({{a}_{1}}=\) first term, \(d=\) common difference (2nd term minus 1st in arithmetic sequence), \(r=\) common ratio (2nd term divided by 1st in geometric sequence)

\({{S}_{n}}=\) sum of first \(n\) terms, \({{S}_{\infty }}=\) sum of infinite number of terms.

Arithmetic:

\({{a}_{n}}={{a}_{1}}+\left( {n-1} \right)d\)

(Recursive: \({{a}_{n}}=a,\,\,{{a}_{n}}={{a}_{{n-1}}}+d\))

Geometric:

\({{a}_{n}}={{a}_{1}}{{\left( r \right)}^{{n-1}}}\)(Recursive: \({{a}_{n}}=a,\,\,{{a}_{n}}=r{{a}_{{n-1}}}\))

\(\displaystyle {{S}_{n}}=\frac{{{{a}_{1}}\left( {1-{{r}^{n}}} \right)}}{{1-r}}\,\,\,\,\,\,{{S}_{\infty }}=\frac{{{{a}_{1}}}}{{1-r}},\,\,\,\,\,\,\,\left| r \right|<1\)

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