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L d 2 q d t 2 + R d q d t + 1 C q = E ( t ) L d 2 q d t 2 + R d q d t + 1 C q = E ( t ) | https://openstax.org/books/calculus-volume-3/pages/7-key-equations |
boundary conditions : the conditions that give the state of a system at different times, such as the position of a spring-mass system at two different times | https://openstax.org/books/calculus-volume-3/pages/7-key-terms |
boundary-value problem : a differential equation with associated boundary conditions | https://openstax.org/books/calculus-volume-3/pages/7-key-terms |
characteristic equation : the equationaλ2+bλ+c=0aλ2+bλ+c=0for the differential equationayâ³+byâ²+cy=0ayâ³+byâ²+cy=0 | https://openstax.org/books/calculus-volume-3/pages/7-key-terms |
complementary equation : for the nonhomogeneous linear differential equationa2(x)yâ³+a1(x)yâ²+a0(x)y=r(x),a2(x)yâ³+a1(x)yâ²+a0(x)y=r(x),the associated homogeneous equation, called thecomplementary equation, isa2(x)yâ³+a1(x)yâ²+a0(x)y=0a2(x)yâ³+a1(x)yâ²+a0(x)y=0 | https://openstax.org/books/calculus-volume-3/pages/7-key-terms |
homogeneous linear equation : a second-order differential equation that can be written in the forma2(x)yâ³+a1(x)yâ²+a0(x)y=r(x),a2(x)yâ³+a1(x)yâ²+a0(x)y=r(x),butr(x)=0r(x)=0for every value ofxx | https://openstax.org/books/calculus-volume-3/pages/7-key-terms |
linearly dependent : a set of functionsf1(x),f2(x),â¦,fn(x)f1(x),f2(x),â¦,fn(x)for which there are constantsc1,c2,â¦cn,c1,c2,â¦cn,not all zero, such thatc1f1(x)+c2f2(x)+â¯+cnfn(x)=0c1f1(x)+c2f2(x)+â¯+cnfn(x)=0for allxin the interval of interest | https://openstax.org/books/calculus-volume-3/pages/7-key-terms |
linearly independent : a set of functionsf1(x),f2(x),â¦,fn(x)f1(x),f2(x),â¦,fn(x)for which there are no constantsc1,c2,â¦cn,c1,c2,â¦cn,such thatc1f1(x)+c2f2(x)+â¯+cnfn(x)=0c1f1(x)+c2f2(x)+â¯+cnfn(x)=0for allxin the interval of interest | https://openstax.org/books/calculus-volume-3/pages/7-key-terms |
method of undetermined coefficients : a method that involves making a guess about the form of the particular solution, then solving for the coefficients in the guess | https://openstax.org/books/calculus-volume-3/pages/7-key-terms |
method of variation of parameters : a method that involves looking for particular solutions in the formyp(x)=u(x)y1(x)+v(x)y2(x),yp(x)=u(x)y1(x)+v(x)y2(x),wherey1y1andy2y2are linearly independent solutions to the complementary equations, and then solving a system of equations to findu(x)u(x)andv(x)v(x) | https://openstax.org/books/calculus-volume-3/pages/7-key-terms |
nonhomogeneous linear equation : a second-order differential equation that can be written in the forma2(x)yâ³+a1(x)yâ²+a0(x)y=r(x),a2(x)yâ³+a1(x)yâ²+a0(x)y=r(x),butr(x)â0r(x)â0for some value ofxx | https://openstax.org/books/calculus-volume-3/pages/7-key-terms |
particular solution : a solutionyp(x)yp(x)of a differential equation that contains no arbitrary constants | https://openstax.org/books/calculus-volume-3/pages/7-key-terms |
RLCseries circuit : a complete electrical path consisting of a resistor, an inductor, and a capacitor; a second-order, constant-coefficient differential equation can be used to model the charge on the capacitor in anRLCseries circuit | https://openstax.org/books/calculus-volume-3/pages/7-key-terms |
simple harmonic motion : motion described by the equationx(t)=c1cos(Ït)+c2sin(Ït),x(t)=c1cos(Ït)+c2sin(Ït),as exhibited by an undamped spring-mass system in which the mass continues to oscillate indefinitely | https://openstax.org/books/calculus-volume-3/pages/7-key-terms |
steady-state solution : a solution to a nonhomogeneous differential equation related to the forcing function; in the long term, the solution approaches the steady-state solution | https://openstax.org/books/calculus-volume-3/pages/7-key-terms |
Rational numbers may be written as fractions or terminating or repeating decimals. SeeExample 1andExample 2. | https://openstax.org/books/college-algebra-2e/pages/1-key-concepts |
Determine whether a number is rational or irrational by writing it as a decimal. SeeExample 3. | https://openstax.org/books/college-algebra-2e/pages/1-key-concepts |
The rational numbers and irrational numbers make up the set of real numbers. SeeExample 4. A number can be classified as natural, whole, integer, rational, or irrational. SeeExample 5. | https://openstax.org/books/college-algebra-2e/pages/1-key-concepts |
The order of operations is used to evaluate expressions. SeeExample 6. | https://openstax.org/books/college-algebra-2e/pages/1-key-concepts |
The real numbers under the operations of addition and multiplication obey basic rules, known as the properties of real numbers. These are the commutative properties, the associative properties, the distributive property, the identity properties, and the inverse properties. SeeExample 7. | https://openstax.org/books/college-algebra-2e/pages/1-key-concepts |
Algebraic expressions are composed of constants and variables that are combined using addition, subtraction, multiplication, and division. SeeExample 8. They take on a numerical value when evaluated by replacing variables with constants. SeeExample 9,Example 10, andExample 12 | https://openstax.org/books/college-algebra-2e/pages/1-key-concepts |
Formulas are equations in which one quantity is represented in terms of other quantities. They may be simplified or evaluated as any mathematical expression. SeeExample 11andExample 13. | https://openstax.org/books/college-algebra-2e/pages/1-key-concepts |
Products of exponential expressions with the same base can be simplified by adding exponents. SeeExample 1. | https://openstax.org/books/college-algebra-2e/pages/1-key-concepts |
Quotients of exponential expressions with the same base can be simplified by subtracting exponents. SeeExample 2. | https://openstax.org/books/college-algebra-2e/pages/1-key-concepts |
Powers of exponential expressions with the same base can be simplified by multiplying exponents. SeeExample 3. | https://openstax.org/books/college-algebra-2e/pages/1-key-concepts |
An expression with exponent zero is defined as 1. SeeExample 4. | https://openstax.org/books/college-algebra-2e/pages/1-key-concepts |
An expression with a negative exponent is defined as a reciprocal. SeeExample 5andExample 6. | https://openstax.org/books/college-algebra-2e/pages/1-key-concepts |
The power of a product of factors is the same as the product of the powers of the same factors. SeeExample 7. | https://openstax.org/books/college-algebra-2e/pages/1-key-concepts |
The power of a quotient of factors is the same as the quotient of the powers of the same factors. SeeExample 8. | https://openstax.org/books/college-algebra-2e/pages/1-key-concepts |
The rules for exponential expressions can be combined to simplify more complicated expressions. SeeExample 9. | https://openstax.org/books/college-algebra-2e/pages/1-key-concepts |
Scientific notation uses powers of 10 to simplify very large or very small numbers. SeeExample 10andExample 11. | https://openstax.org/books/college-algebra-2e/pages/1-key-concepts |
Scientific notation may be used to simplify calculations with very large or very small numbers. SeeExample 12andExample 13. | https://openstax.org/books/college-algebra-2e/pages/1-key-concepts |
The principal square root of a numberaais the nonnegative number that when multiplied by itself equalsa.a.SeeExample 1. | https://openstax.org/books/college-algebra-2e/pages/1-key-concepts |
Ifaaandbbare nonnegative, the square root of the productababis equal to the product of the square roots ofaaandbbSeeExample 2andExample 3. | https://openstax.org/books/college-algebra-2e/pages/1-key-concepts |
Ifaaandbbare nonnegative, the square root of the quotientababis equal to the quotient of the square roots ofaaandbbSeeExample 4andExample 5. | https://openstax.org/books/college-algebra-2e/pages/1-key-concepts |
We can add and subtract radical expressions if they have the same radicand and the same index. SeeExample 6andExample 7. | https://openstax.org/books/college-algebra-2e/pages/1-key-concepts |
Radical expressions written in simplest form do not contain a radical in the denominator. To eliminate the square root radical from the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. SeeExample 8andExample 9. | https://openstax.org/books/college-algebra-2e/pages/1-key-concepts |
The principalnth root ofaais the number with the same sign asaathat when raised to thenth power equalsa.a.These roots have the same properties as square roots.SeeExample 10. | https://openstax.org/books/college-algebra-2e/pages/1-key-concepts |
Radicals can be rewritten as rational exponents and rational exponents can be rewritten as radicals. SeeExample 11andExample 12. | https://openstax.org/books/college-algebra-2e/pages/1-key-concepts |
The properties of exponents apply to rational exponents. SeeExample 13. | https://openstax.org/books/college-algebra-2e/pages/1-key-concepts |
A polynomial is a sum of terms each consisting of a variable raised to a non-negative integer power. The degree is the highest power of the variable that occurs in the polynomial. The leading term is the term containing the highest degree, and the leading coefficient is the coefficient of that term. SeeExample 1. | https://openstax.org/books/college-algebra-2e/pages/1-key-concepts |
We can add and subtract polynomials by combining like terms. SeeExample 2andExample 3. | https://openstax.org/books/college-algebra-2e/pages/1-key-concepts |
To multiply polynomials, use the distributive property to multiply each term in the first polynomial by each term in the second. Then add the products. SeeExample 4. | https://openstax.org/books/college-algebra-2e/pages/1-key-concepts |
FOIL (First, Outer, Inner, Last) is a shortcut that can be used to multiply binomials. SeeExample 5. | https://openstax.org/books/college-algebra-2e/pages/1-key-concepts |
Perfect square trinomials and difference of squares are special products. SeeExample 6andExample 7. | https://openstax.org/books/college-algebra-2e/pages/1-key-concepts |
Follow the same rules to work with polynomials containing several variables. SeeExample 8. | https://openstax.org/books/college-algebra-2e/pages/1-key-concepts |
The greatest common factor, or GCF, can be factored out of a polynomial. Checking for a GCF should be the first step in any factoring problem. SeeExample 1. | https://openstax.org/books/college-algebra-2e/pages/1-key-concepts |
Trinomials with leading coefficient 1 can be factored by finding numbers that have a product of the third term and a sum of the second term. SeeExample 2. | https://openstax.org/books/college-algebra-2e/pages/1-key-concepts |
Trinomials can be factored using a process called factoring by grouping. SeeExample 3. | https://openstax.org/books/college-algebra-2e/pages/1-key-concepts |
Perfect square trinomials and the difference of squares are special products and can be factored using equations. SeeExample 4andExample 5. | https://openstax.org/books/college-algebra-2e/pages/1-key-concepts |
The sum of cubes and the difference of cubes can be factored using equations. SeeExample 6andExample 7. | https://openstax.org/books/college-algebra-2e/pages/1-key-concepts |
Polynomials containing fractional and negative exponents can be factored by pulling out a GCF. SeeExample 8. | https://openstax.org/books/college-algebra-2e/pages/1-key-concepts |
Rational expressions can be simplified by cancelling common factors in the numerator and denominator. SeeExample 1. | https://openstax.org/books/college-algebra-2e/pages/1-key-concepts |
We can multiply rational expressions by multiplying the numerators and multiplying the denominators. SeeExample 2. | https://openstax.org/books/college-algebra-2e/pages/1-key-concepts |
To divide rational expressions, multiply by the reciprocal of the second expression. SeeExample 3. | https://openstax.org/books/college-algebra-2e/pages/1-key-concepts |
Adding or subtracting rational expressions requires finding a common denominator. SeeExample 4andExample 5. | https://openstax.org/books/college-algebra-2e/pages/1-key-concepts |
Complex rational expressions have fractions in the numerator or the denominator. These expressions can be simplified. SeeExample 6. | https://openstax.org/books/college-algebra-2e/pages/1-key-concepts |
a m â a n = a m + n a m â a n = a m + n | https://openstax.org/books/college-algebra-2e/pages/1-key-equations |
a m a n = a m â n a m a n = a m â n | https://openstax.org/books/college-algebra-2e/pages/1-key-equations |
( a m ) n = a m â n ( a m ) n = a m â n | https://openstax.org/books/college-algebra-2e/pages/1-key-equations |
a 0 = 1 a 0 = 1 | https://openstax.org/books/college-algebra-2e/pages/1-key-equations |
a â n = 1 a n a â n = 1 a n | https://openstax.org/books/college-algebra-2e/pages/1-key-equations |
( a â b ) n = a n â b n ( a â b ) n = a n â b n | https://openstax.org/books/college-algebra-2e/pages/1-key-equations |
( a b ) n = a n b n ( a b ) n = a n b n | https://openstax.org/books/college-algebra-2e/pages/1-key-equations |
( x + a ) 2 = ( x + a ) ( x + a ) = x 2 + 2 a x + a 2 ( x + a ) 2 = ( x + a ) ( x + a ) = x 2 + 2 a x + a 2 | https://openstax.org/books/college-algebra-2e/pages/1-key-equations |
( a + b ) ( a â b ) = a 2 â b 2 ( a + b ) ( a â b ) = a 2 â b 2 | https://openstax.org/books/college-algebra-2e/pages/1-key-equations |
a 2 â b 2 = ( a + b ) ( a â b ) a 2 â b 2 = ( a + b ) ( a â b ) | https://openstax.org/books/college-algebra-2e/pages/1-key-equations |
a 2 + 2 a b + b 2 = ( a + b ) 2 a 2 + 2 a b + b 2 = ( a + b ) 2 | https://openstax.org/books/college-algebra-2e/pages/1-key-equations |
a 3 + b 3 = ( a + b ) ( a 2 â a b + b 2 ) a 3 + b 3 = ( a + b ) ( a 2 â a b + b 2 ) | https://openstax.org/books/college-algebra-2e/pages/1-key-equations |
a 3 â b 3 = ( a â b ) ( a 2 + a b + b 2 ) a 3 â b 3 = ( a â b ) ( a 2 + a b + b 2 ) | https://openstax.org/books/college-algebra-2e/pages/1-key-equations |
algebraic expression : constants and variables combined using addition, subtraction, multiplication, and division | https://openstax.org/books/college-algebra-2e/pages/1-key-terms |
associative property of addition : the sum of three numbers may be grouped differently without affecting the result; in symbols,a+(b+c)=(a+b)+ca+(b+c)=(a+b)+c | https://openstax.org/books/college-algebra-2e/pages/1-key-terms |
associative property of multiplication : the product of three numbers may be grouped differently without affecting the result; in symbols,aâ(bâc)=(aâb)âcaâ(bâc)=(aâb)âc | https://openstax.org/books/college-algebra-2e/pages/1-key-terms |
base : in exponential notation, the expression that is being multiplied | https://openstax.org/books/college-algebra-2e/pages/1-key-terms |
binomial : a polynomial containing two terms | https://openstax.org/books/college-algebra-2e/pages/1-key-terms |
coefficient : any real numberaiaiin a polynomial in the formanxn+...+a2x2+a1x+a0anxn+...+a2x2+a1x+a0 | https://openstax.org/books/college-algebra-2e/pages/1-key-terms |
commutative property of addition : two numbers may be added in either order without affecting the result; in symbols,a+b=b+aa+b=b+a | https://openstax.org/books/college-algebra-2e/pages/1-key-terms |
commutative property of multiplication : two numbers may be multiplied in any order without affecting the result; in symbols,aâb=bâaaâb=bâa | https://openstax.org/books/college-algebra-2e/pages/1-key-terms |
constant : a quantity that does not change value | https://openstax.org/books/college-algebra-2e/pages/1-key-terms |
degree : the highest power of the variable that occurs in a polynomial | https://openstax.org/books/college-algebra-2e/pages/1-key-terms |
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/college-algebra-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/college-algebra-2e/pages/1-key-terms |
equation : a mathematical statement indicating that two expressions are equal | https://openstax.org/books/college-algebra-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/college-algebra-2e/pages/1-key-terms |
exponential notation : a shorthand method of writing products of the same factor | https://openstax.org/books/college-algebra-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/college-algebra-2e/pages/1-key-terms |
formula : an equation expressing a relationship between constant and variable quantities | https://openstax.org/books/college-algebra-2e/pages/1-key-terms |
greatest common factor : the largest polynomial that divides evenly into each polynomial | https://openstax.org/books/college-algebra-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/college-algebra-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/college-algebra-2e/pages/1-key-terms |
index : the number above the radical sign indicating thenth root | https://openstax.org/books/college-algebra-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/college-algebra-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/college-algebra-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/college-algebra-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/college-algebra-2e/pages/1-key-terms |
leading coefficient : the coefficient of the leading term | https://openstax.org/books/college-algebra-2e/pages/1-key-terms |
leading term : the term containing the highest degree | https://openstax.org/books/college-algebra-2e/pages/1-key-terms |
least common denominator : the smallest multiple that two denominators have in common | https://openstax.org/books/college-algebra-2e/pages/1-key-terms |
monomial : a polynomial containing one term | https://openstax.org/books/college-algebra-2e/pages/1-key-terms |
natural numbers : the set of counting numbers:{1,2,3,â¦}{1,2,3,â¦} | https://openstax.org/books/college-algebra-2e/pages/1-key-terms |
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