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difference of squares : the binomial that results when a binomial is multiplied by a binomial with the same terms, but the opposite sign | https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-key-terms |
distributive property : the product of a factor times a sum is the sum of the factor times each term in the sum; in symbols,aâ(b+c)=aâb+aâcaâ(b+c)=aâb+aâc | https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-key-terms |
equation : a mathematical statement indicating that two expressions are equal | https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-key-terms |
exponent : in exponential notation, the raised number or variable that indicates how many times the base is being multiplied | https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-key-terms |
exponential notation : a shorthand method of writing products of the same factor | https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-key-terms |
factor by grouping : a method for factoring a trinomial in the formax2+bx+cax2+bx+cby dividing thexterm into the sum of two terms, factoring each portion of the expression separately, and then factoring out the GCF of the entire expression | https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-key-terms |
formula : an equation expressing a relationship between constant and variable quantities | https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-key-terms |
greatest common factor : the largest polynomial that divides evenly into each polynomial | https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-key-terms |
identity property of addition : there is a unique number, called the additive identity, 0, which, when added to a number, results in the original number; in symbols,a+0=aa+0=a | https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-key-terms |
identity property of multiplication : there is a unique number, called the multiplicative identity, 1, which, when multiplied by a number, results in the original number; in symbols,aâ1=aaâ1=a | https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-key-terms |
index : the number above the radical sign indicating thenth root | https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-key-terms |
integers : the set consisting of the natural numbers, their opposites, and 0:{â¦,â3,â2,â1,0,1,2,3,â¦}{â¦,â3,â2,â1,0,1,2,3,â¦} | https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-key-terms |
inverse property of addition : for every real numbera,a,there is a unique number, called the additive inverse (or opposite), denotedâa,âa,which, when added to the original number, results in the additive identity, 0; in symbols,a+(âa)=0a+(âa)=0 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-key-terms |
inverse property of multiplication : for every non-zero real numbera,a,there is a unique number, called the multiplicative inverse (or reciprocal), denoted1a,1a,which, when multiplied by the original number, results in the multiplicative identity, 1; in symbols,aâ1a=1aâ1a=1 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-key-terms |
irrational numbers : the set of all numbers that are not rational; they cannot be written as either a terminating or repeating decimal; they cannot be expressed as a fraction of two integers | https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-key-terms |
leading coefficient : the coefficient of the leading term | https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-key-terms |
leading term : the term containing the highest degree | https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-key-terms |
least common denominator : the smallest multiple that two denominators have in common | https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-key-terms |
monomial : a polynomial containing one term | https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-key-terms |
natural numbers : the set of counting numbers:{1,2,3,â¦}{1,2,3,â¦} | https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-key-terms |
order of operations : a set of rules governing how mathematical expressions are to be evaluated, assigning priorities to operations | https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-key-terms |
perfect square trinomial : the trinomial that results when a binomial is squared | https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-key-terms |
polynomial : a sum of terms each consisting of a variable raised to a nonnegative integer power | https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-key-terms |
principalnth root : the number with the same sign asaathat when raised to thenth power equalsaa | https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-key-terms |
principal square root : the nonnegative square root of a numberaathat, when multiplied by itself, equalsaa | https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-key-terms |
radical : the symbol used to indicate a root | https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-key-terms |
radical expression : an expression containing a radical symbol | https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-key-terms |
radicand : the number under the radical symbol | https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-key-terms |
rational expression : the quotient of two polynomial expressions | https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-key-terms |
rational numbers : the set of all numbers of the formmn,mn,wheremmandnnare integers andnâ0.nâ0.Any rational number may be written as a fraction or a terminating or repeating decimal. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-key-terms |
real number line : a horizontal line used to represent the real numbers. An arbitrary fixed point is chosen to represent 0; positive numbers lie to the right of 0 and negative numbers to the left. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-key-terms |
real numbers : the sets of rational numbers and irrational numbers taken together | https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-key-terms |
scientific notation : a shorthand notation for writing very large or very small numbers in the formaÃ10naÃ10nwhere1â¤|a|<101â¤|a|<10andnnis an integer | https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-key-terms |
term of a polynomial : anyaixiaixiof a polynomial in the formanxn+...+a2x2+a1x+a0anxn+...+a2x2+a1x+a0 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-key-terms |
trinomial : a polynomial containing three terms | https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-key-terms |
variable : a quantity that may change value | https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-key-terms |
whole numbers : the set consisting of 0 plus the natural numbers:{0,1,2,3,â¦}{0,1,2,3,â¦} | https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-key-terms |
We can locate, or plot, points in the Cartesian coordinate system using ordered pairs, which are defined as displacement from thex-axis and displacement from they-axis. SeeExample 1. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-concepts |
An equation can be graphed in the plane by creating a table of values and plotting points. SeeExample 2. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-concepts |
Using a graphing calculator or a computer program makes graphing equations faster and more accurate. Equations usually have to be entered in the formy=_____. SeeExample 3. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-concepts |
Finding thex-andy-intercepts can define the graph of a line. These are the points where the graph crosses the axes. SeeExample 4. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-concepts |
The distance formula is derived from the Pythagorean Theorem and is used to find the length of a line segment. SeeExample 5andExample 6. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-concepts |
The midpoint formula provides a method of finding the coordinates of the midpoint dividing the sum of thex-coordinates and the sum of they-coordinates of the endpoints by 2. SeeExample 7andExample 8. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-concepts |
We can solve linear equations in one variable in the formax+b=0ax+b=0using standard algebraic properties. SeeExample 1andExample 2. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-concepts |
A rational expression is a quotient of two polynomials. We use the LCD to clear the fractions from an equation. SeeExample 3andExample 4. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-concepts |
All solutions to a rational equation should be verified within the original equation to avoid an undefined term, or zero in the denominator. SeeExample 5andExample 6andExample 7. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-concepts |
Given two points, we can find the slope of a line using the slope formula. SeeExample 8. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-concepts |
We can identify the slope andy-intercept of an equation in slope-intercept form. SeeExample 9. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-concepts |
We can find the equation of a line given the slope and a point. SeeExample 10. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-concepts |
We can also find the equation of a line given two points. Find the slope and use the point-slope formula. SeeExample 11. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-concepts |
The standard form of a line has no fractions. SeeExample 12. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-concepts |
Horizontal lines have a slope of zero and are defined asy=c,y=c,wherecis a constant. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-concepts |
Vertical lines have an undefined slope (zero in the denominator), and are defined asx=c,x=c,wherecis a constant. SeeExample 13. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-concepts |
Parallel lines have the same slope and differenty-intercepts. SeeExample 14andExample 15. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-concepts |
Perpendicular lines have slopes that are negative reciprocals of each other unless one is horizontal and the other is vertical. SeeExample 16. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-concepts |
A linear equation can be used to solve for an unknown in a number problem. SeeExample 1. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-concepts |
Applications can be written as mathematical problems by identifying known quantities and assigning a variable to unknown quantities. SeeExample 2. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-concepts |
There are many known formulas that can be used to solve applications. Distance problems, for example, are solved using thed=rtd=rtformula. SeeExample 3. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-concepts |
Many geometry problems are solved using the perimeter formulaP=2L+2W,P=2L+2W,the area formulaA=LW,A=LW,or the volume formulaV=LWH.V=LWH.SeeExample 4,Example 5, andExample 6. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-concepts |
The square root of any negative number can be written as a multiple ofi.i.SeeExample 1. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-concepts |
To plot a complex number, we use two number lines, crossed to form the complex plane. The horizontal axis is the real axis, and the vertical axis is the imaginary axis. SeeExample 2. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-concepts |
Complex numbers can be added and subtracted by combining the real parts and combining the imaginary parts. SeeExample 3. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-concepts |
Complex numbers can be multiplied and divided.To multiply complex numbers, distribute just as with polynomials. SeeExample 4andExample 5.To divide complex numbers, multiply both numerator and denominator by the complex conjugate of the denominator to eliminate the complex number from the denominator. SeeExample 6andExa... | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-concepts |
To multiply complex numbers, distribute just as with polynomials. SeeExample 4andExample 5. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-concepts |
To divide complex numbers, multiply both numerator and denominator by the complex conjugate of the denominator to eliminate the complex number from the denominator. SeeExample 6andExample 7. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-concepts |
The powers ofiiare cyclic, repeating every fourth one. SeeExample 8. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-concepts |
Many quadratic equations can be solved by factoring when the equation has a leading coefficient of 1 or if the equation is a difference of squares. The zero-product property is then used to find solutions. SeeExample 1,Example 2, andExample 3. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-concepts |
Many quadratic equations with a leading coefficient other than 1 can be solved by factoring using the grouping method. SeeExample 4andExample 5. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-concepts |
Another method for solving quadratics is the square root property. The variable is squared. We isolate the squared term and take the square root of both sides of the equation. The solution will yield a positive and negative solution. SeeExample 6andExample 7. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-concepts |
Completing the square is a method of solving quadratic equations when the equation cannot be factored. SeeExample 8. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-concepts |
A highly dependable method for solving quadratic equations is the quadratic formula, based on the coefficients and the constant term in the equation. SeeExample 9andExample 10. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-concepts |
The discriminant is used to indicate the nature of the roots that the quadratic equation will yield: real or complex, rational or irrational, and how many of each. SeeExample 11. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-concepts |
The Pythagorean Theorem, among the most famous theorems in history, is used to solve right-triangle problems and has applications in numerous fields. Solving for the length of one side of a right triangle requires solving a quadratic equation. SeeExample 12. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-concepts |
Rational exponents can be rewritten several ways depending on what is most convenient for the problem. To solve, both sides of the equation are raised to a power that will render the exponent on the variable equal to 1. SeeExample 1,Example 2, andExample 3. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-concepts |
Factoring extends to higher-order polynomials when it involves factoring out the GCF or factoring by grouping. SeeExample 4andExample 5. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-concepts |
We can solve radical equations by isolating the radical and raising both sides of the equation to a power that matches the index. SeeExample 6andExample 7. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-concepts |
To solve absolute value equations, we need to write two equations, one for the positive value and one for the negative value. SeeExample 8. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-concepts |
Equations in quadratic form are easy to spot, as the exponent on the first term is double the exponent on the second term and the third term is a constant. We may also see a binomial in place of the single variable. We use substitution to solve. SeeExample 9andExample 10. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-concepts |
Solving a rational equation may also lead to a quadratic equation or an equation in quadratic form. SeeExample 11. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-concepts |
Interval notation is a method to indicate the solution set to an inequality. Highly applicable in calculus, it is a system of parentheses and brackets that indicate what numbers are included in a set and whether the endpoints are included as well. SeeTable 1andExample 2. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-concepts |
Solving inequalities is similar to solving equations. The same algebraic rules apply, except for one: multiplying or dividing by a negative number reverses the inequality. SeeExample 3,Example 4,Example 5, andExample 6. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-concepts |
Compound inequalities often have three parts and can be rewritten as two independent inequalities. Solutions are given by boundary values, which are indicated as a beginning boundary or an ending boundary in the solutions to the two inequalities. SeeExample 7andExample 8. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-concepts |
Absolute value inequalities will produce two solution sets due to the nature of absolute value. We solve by writing two equations: one equal to a positive value and one equal to a negative value. SeeExample 9andExample 10. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-concepts |
Absolute value inequalities can also be solved by graphing. At least we can check the algebraic solutions by graphing, as we cannot depend on a visual for a precise solution. SeeExample 11. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-concepts |
x = â b ± b 2 â 4 a c 2 a x = â b ± b 2 â 4 a c 2 a | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-equations |
d = x 2 - x 1 2 + y 2 - y 1 2 d = x 2 - x 1 2 + y 2 - y 1 2 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-equations |
m = y 2 - y 1 x 2 - x 1 m = y 2 - y 1 x 2 - x 1 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-equations |
y - y 1 = m x - x 1 y - y 1 = m x - x 1 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-equations |
( a + b i ) + ( c + d i ) = ( a + c ) + ( b + d ) i ( a + b i ) + ( c + d i ) = ( a + c ) + ( b + d ) i | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-equations |
( a + b i ) - ( c + d i ) = ( a - c ) + ( b - d ) i ( a + b i ) - ( c + d i ) = ( a - c ) + ( b - d ) i | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-equations |
If x 2 = k then x = ± k If x 2 = k then x = ± k | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-equations |
If a x 2 + b x + c = 0 then x = - b ± b 2 - 4 a c 2 a If a x 2 + b x + c = 0 then x = - b ± b 2 - 4 a c 2 a | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-equations |
a m n = a 1 n m = a m 1 m = a m n = ( a n ) m a m n = a 1 n m = a m 1 m = a m n = ( a n ) m | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-equations |
If x ⥠0 , | x | = x ; If x < 0 , | x | = - x If x ⥠0 , | x | = x ; If x < 0 , | x | = - x | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-equations |
If a < b then a + c < b + c If a < b then a + c < b + c | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-equations |
absolute value equation : an equation in which the variable appears in absolute value bars, typically with two solutions, one accounting for the positive expression and one for the negative expression | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-terms |
area : in square units, the area formula used in this section is used to find the area of any two-dimensional rectangular region:A=LWA=LW | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-terms |
Cartesian coordinate system : a grid system designed with perpendicular axes invented by René Descartes | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-terms |
completing the square : a process for solving quadratic equations in which terms are added to or subtracted from both sides of the equation in order to make one side a perfect square | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-terms |
complex conjugate : a complex number containing the same terms as another complex number, but with the opposite operator. Multiplying a complex number by its conjugate yields a real number. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-key-terms |
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