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Solutions of a System of Linear Inequalities:Solutions of a system of linear inequalities are the values of the variables that make all the inequalities true. The solution of a system of linear inequalities is shown as a shaded region in thex, ycoordinate system that includes all the points whose ordered pairs make the... | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
How to solve a system of linear inequalities by graphing.Step 1.Graph the first inequality.Graph the boundary line.Shade in the side of the boundary line where the inequality is true.Step 2.On the same grid, graph the second inequality.Graph the boundary line.Shade in the side of that boundary line where the inequality... | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 1.Graph the first inequality.Graph the boundary line.Shade in the side of the boundary line where the inequality is true. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 2.On the same grid, graph the second inequality.Graph the boundary line.Shade in the side of that boundary line where the inequality is true. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 3.The solution is the region where the shading overlaps. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 4.Check by choosing a test point. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
break-even point : The point at which the revenue equals the costs is the break-even point;C(x)=R(x).C(x)=R(x). | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-terms |
coincident lines : Coincident lines have the same slope and samey-intercept. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-terms |
complementary angles : Two angles are complementary if the sum of the measures of their angles is 90 degrees. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-terms |
consistent and inconsistent systems : Consistent system of equations is a system of equations with at least one solution; inconsistent system of equations is a system of equations with no solution. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-terms |
cost function : The cost function is the cost to manufacture each unit timesxx, the number of units manufactured, plus the fixed costs;C(x) = (cost per unit)x+ fixed costs. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-terms |
determinant : Each square matrix has a real number associated with it called its determinant. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-terms |
matrix : A matrix is a rectangular array of numbers arranged in rows and columns. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-terms |
minor of an entry in a3Ã33Ã3determinant : The minor of an entry in a3Ã33Ã3determinant is the2Ã22Ã2determinant found by eliminating the row and column in the3Ã33Ã3determinant that contains the entry. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-terms |
revenue : The revenue is the selling price of each unit timesx, the number of units sold;R(x) = (selling price per unit)x. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-terms |
row-echelon form : A matrix is in row-echelon form when to the left of the vertical line, each entry on the diagonal is a 1 and all entries below the diagonal are zeros. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-terms |
solutions of a system of equations : Solutions of a system of equations are the values of the variables that makeallthe equations true; solution is represented by an ordered pair(x,y).(x,y). | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-terms |
solutions of a system of linear equations with three variables : The solutions of a system of equations are the values of the variables that make all the equations true; a solution is represented by an ordered triple(x,y,z).(x,y,z). | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-terms |
square matrix : A square matrix is a matrix with the same number of rows and columns. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-terms |
supplementary angles : Two angles are supplementary if the sum of the measures of their angles is 180 degrees. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-terms |
system of linear equations : When two or more linear equations are grouped together, they form a system of linear equations. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-terms |
system of linear inequalities : Two or more linear inequalities grouped together form a system of linear inequalities. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-terms |
a m · a n = a m + n a m · a n = a m + n | https://openstax.org/books/intermediate-algebra-2e/pages/5-key-concepts |
( a m ) n = a m · n ( a m ) n = a m · n | https://openstax.org/books/intermediate-algebra-2e/pages/5-key-concepts |
( a b ) n = a n b n ( a b ) n = a n b n | https://openstax.org/books/intermediate-algebra-2e/pages/5-key-concepts |
a m a n = a m â n , a â 0 a m a n = a m â n , a â 0 | https://openstax.org/books/intermediate-algebra-2e/pages/5-key-concepts |
a 0 = 1 , a â 0 a 0 = 1 , a â 0 | https://openstax.org/books/intermediate-algebra-2e/pages/5-key-concepts |
( a b ) m = a m b m , b â 0 ( a b ) m = a m b m , b â 0 | https://openstax.org/books/intermediate-algebra-2e/pages/5-key-concepts |
a â n = 1 a n a â n = 1 a n and 1 a â n = a n 1 a â n = a n | https://openstax.org/books/intermediate-algebra-2e/pages/5-key-concepts |
( a b ) â n = ( b a ) n ( a b ) â n = ( b a ) n | https://openstax.org/books/intermediate-algebra-2e/pages/5-key-concepts |
( a + b ) 2 = a 2 + 2 a b + b 2 ( a + b ) 2 = a 2 + 2 a b + b 2 | https://openstax.org/books/intermediate-algebra-2e/pages/5-key-concepts |
( a â b ) ( a + b ) = a 2 â b 2 ( a â b ) ( a + b ) = a 2 â b 2 | https://openstax.org/books/intermediate-algebra-2e/pages/5-key-concepts |
( a â b ) 2 = a 2 â 2 a b + b 2 ( a â b ) 2 = a 2 â 2 a b + b 2 | https://openstax.org/books/intermediate-algebra-2e/pages/5-key-concepts |
binomial : A binomial is a polynomial with exactly two terms. | https://openstax.org/books/intermediate-algebra-2e/pages/5-key-terms |
conjugate pair : A conjugate pair is two binomials of the form(aâb),(a+b).(aâb),(a+b).The pair of binomials each have the same first term and the same last term, but one binomial is a sum and the other is a difference. | https://openstax.org/books/intermediate-algebra-2e/pages/5-key-terms |
degree of a constant : The degree of any constant is 0. | https://openstax.org/books/intermediate-algebra-2e/pages/5-key-terms |
degree of a polynomial : The degree of a polynomial is the highest degree of all its terms. | https://openstax.org/books/intermediate-algebra-2e/pages/5-key-terms |
degree of a term : The degree of a term is the sum of the exponents of its variables. | https://openstax.org/books/intermediate-algebra-2e/pages/5-key-terms |
monomial : A monomial is an algebraic expression with one term. A monomial in one variable is a term of the formaxm,axm,whereais a constant andmis a whole number. | https://openstax.org/books/intermediate-algebra-2e/pages/5-key-terms |
polynomial : A monomial or two or more monomials combined by addition or subtraction is a polynomial. | https://openstax.org/books/intermediate-algebra-2e/pages/5-key-terms |
polynomial function : A polynomial function is a function whose range values are defined by a polynomial. | https://openstax.org/books/intermediate-algebra-2e/pages/5-key-terms |
Power Property : According to the Power Property,ato themto thenequalsato themtimesn. | https://openstax.org/books/intermediate-algebra-2e/pages/5-key-terms |
Product Property : According to the Product Property,ato themtimesato thenequalsato themplusn. | https://openstax.org/books/intermediate-algebra-2e/pages/5-key-terms |
Product to a Power : According to the Product to a Power Property,atimesbin parentheses to themequalsato themtimesbto them. | https://openstax.org/books/intermediate-algebra-2e/pages/5-key-terms |
Properties of Negative Exponents : According to the Properties of Negative Exponents,ato the negativenequals 1 divided byato thenand 1 divided byato the negativenequalsato then. | https://openstax.org/books/intermediate-algebra-2e/pages/5-key-terms |
Quotient Property : According to the Quotient Property,ato themdivided byato thenequalsato themminusnas long asais not zero. | https://openstax.org/books/intermediate-algebra-2e/pages/5-key-terms |
Quotient to a Negative Exponent : Raising a quotient to a negative exponent occurs whenadivided bybin parentheses to the power of negativenequalsbdivided byain parentheses to the power ofn. | https://openstax.org/books/intermediate-algebra-2e/pages/5-key-terms |
Quotient to a Power Property : According to the Quotient to a Power Property,adivided bybin parentheses to the power ofmis equal toato themdivided bybto themas long asbis not zero. | https://openstax.org/books/intermediate-algebra-2e/pages/5-key-terms |
standard form of a polynomial : A polynomial is in standard form when the terms of a polynomial are written in descending order of degrees. | https://openstax.org/books/intermediate-algebra-2e/pages/5-key-terms |
trinomial : A trinomial is a polynomial with exactly three terms. | https://openstax.org/books/intermediate-algebra-2e/pages/5-key-terms |
Zero Exponent Property : According to the Zero Exponent Property,ato the zero is 1 as long asais not zero. | https://openstax.org/books/intermediate-algebra-2e/pages/5-key-terms |
How to find the greatest common factor (GCF) of two expressions.Step 1.Factor each coefficient into primes. Write all variables with exponents in expanded form.Step 2.List all factorsâmatching common factors in a column. In each column, circle the common factors.Step 3.Bring down the common factors that all expressio... | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 1.Factor each coefficient into primes. Write all variables with exponents in expanded form. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 2.List all factorsâmatching common factors in a column. In each column, circle the common factors. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 3.Bring down the common factors that all expressions share. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 4.Multiply the factors. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Distributive Property:Ifa,b, andcare real numbers, thena(b+c)=ab+acandab+ac=a(b+c)a(b+c)=ab+acandab+ac=a(b+c)The form on the left is used to multiply. The form on the right is used to factor. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
How to factor the greatest common factor from a polynomial.Step 1.Find the GCF of all the terms of the polynomial.Step 2.Rewrite each term as a product using the GCF.Step 3.Use the âreverseâ Distributive Property to factor the expression.Step 4.Check by multiplying the factors. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 1.Find the GCF of all the terms of the polynomial. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 2.Rewrite each term as a product using the GCF. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 3.Use the âreverseâ Distributive Property to factor the expression. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 4.Check by multiplying the factors. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Factor as a Noun and a Verb:We use âfactorâ as both a noun and a verb.Noun:7 is afactorof 14Verb:factor3 from3a+3Noun:7 is afactorof 14Verb:factor3 from3a+3 | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
How to factor by grouping.Step 1.Group terms with common factors.Step 2.Factor out the common factor in each group.Step 3.Factor the common factor from the expression.Step 4.Check by multiplying the factors. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 1.Group terms with common factors. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 2.Factor out the common factor in each group. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 3.Factor the common factor from the expression. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 4.Check by multiplying the factors. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
How to factor trinomials of the formx2+bx+c.x2+bx+c.Step 1.Write the factors as two binomials with first termsx.x2+bx+c(x)(x)x2+bx+c(x)(x)Step 2.Find two numbersmandnthatmultiply toc,m·n=cadd tob,m+n=bmultiply toc,m·n=cadd tob,m+n=bStep 3.Usemandnas the last terms of the factors.(x+m)(x+n)(x+m)(x+n)Step 4.Check by mu... | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 1.Write the factors as two binomials with first termsx.x2+bx+c(x)(x)x2+bx+c(x)(x) | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 2.Find two numbersmandnthatmultiply toc,m·n=cadd tob,m+n=bmultiply toc,m·n=cadd tob,m+n=b | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 3.Usemandnas the last terms of the factors.(x+m)(x+n)(x+m)(x+n) | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 4.Check by multiplying the factors. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Strategy for Factoring Trinomials of the Formx2+bx+cx2+bx+c:When we factor a trinomial, we look at the signs of its terms first to determine the signs of the binomial factors.x2+bx+c(x+m)(x+n)Whencis positive,mandnhave the same sign.bpositivebnegativem,npositivem,nnegativex2+5x+6x2â6x+8(x+2)(x+3)(xâ4)(xâ2)same si... | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
How to factor trinomials of the formax2+bx+cax2+bx+cusing trial and error.Step 1.Write the trinomial in descending order of degrees as needed.Step 2.Factor any GCF.Step 3.Find all the factor pairs of the first term.Step 4.Find all the factor pairs of the third term.Step 5.Test all the possible combinations of the facto... | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 1.Write the trinomial in descending order of degrees as needed. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 2.Factor any GCF. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 3.Find all the factor pairs of the first term. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 4.Find all the factor pairs of the third term. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 5.Test all the possible combinations of the factors until the correct product is found. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 6.Check by multiplying. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
How to factor trinomials of the formax2+bx+cax2+bx+cusing the âacâ method.Step 1.Factor any GCF.Step 2.Find the productac.Step 3.Find two numbersmandnthat:Multiply toac.m·n=a·cAdd tob.m+n=bax2+bx+cMultiply toac.m·n=a·cAdd tob.m+n=bax2+bx+cStep 4.Split the middle term usingmandn.ax2+mx+nx+cax2+mx+nx+cStep 5.Fact... | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 1.Factor any GCF. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 2.Find the productac. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 3.Find two numbersmandnthat:Multiply toac.m·n=a·cAdd tob.m+n=bax2+bx+cMultiply toac.m·n=a·cAdd tob.m+n=bax2+bx+c | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 4.Split the middle term usingmandn.ax2+mx+nx+cax2+mx+nx+c | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 5.Factor by grouping. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 6.Check by multiplying the factors. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Perfect Square Trinomials Pattern:Ifaandbare real numbers,a2+2ab+b2=(a+b)2a2â2ab+b2=(aâb)2a2+2ab+b2=(a+b)2a2â2ab+b2=(aâb)2 | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
How to factor perfect square trinomials.Step 1.Does the trinomial fit the pattern?a2+2ab+b2a2â2ab+b2Is the first term a perfect square?(a)2(a)2Write it as a square.Is the last term a perfect square?(a)2(b)2(a)2(b)2Write it as a square.Check the middle term. Is it2ab?(a)2â2·a·bâ(b)2(a)2â2·a·bâ(b)2Step 2.Wr... | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Difference of Squares Pattern:Ifa,ba,bare real numbers, | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
How to factor differences of squares.Step 1.Does the binomial fit the pattern?a2âb2Is this a difference?____â____Are the first and last terms perfect squares?Step 2.Write them as squares.(a)2â(b)2Step 3.Write the product of conjugates.(aâb)(a+b)Step 4.Check by multiplying.Step 1.Does the binomial fit the patter... | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Sum and Difference of Cubes Patterna3+b3=(a+b)(a2âab+b2)a3âb3=(aâb)(a2+ab+b2)a3+b3=(a+b)(a2âab+b2)a3âb3=(aâb)(a2+ab+b2) | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
How to factor the sum or difference of cubes.Step 1.Does the binomial fit the sum or difference of cubes pattern?Is it a sum or difference?Are the first and last terms perfect cubes?Step 2.Write them as cubes.Step 3.Use either the sum or difference of cubes pattern.Step 4.Simplify inside the parenthesesStep 5.Check by ... | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 1.Does the binomial fit the sum or difference of cubes pattern?Is it a sum or difference?Are the first and last terms perfect cubes? | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 2.Write them as cubes. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 3.Use either the sum or difference of cubes pattern. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 4.Simplify inside the parentheses | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 5.Check by multiplying the factors. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
How to use a general strategy for factoring polynomials.Step 1.Is there a greatest common factor?Factor it out.Step 2.Is the polynomial a binomial, trinomial, or are there more than three terms?If it is a binomial:Is it a sum?Of squares? Sums of squares do not factor.Of cubes? Use the sum of cubes pattern.Is it a diffe... | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
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