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Step 1.Is there a greatest common factor?Factor it out. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
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 difference?Of squares? Factor as the product of conjugates.Of cubes? Use the difference of cubes pattern.If it is a ... | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 3.Check.Is it factored completely?Do the factors multiply back to the original polynomial? | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Polynomial Equation:A polynomial equation is an equation that contains a polynomial expression. The degree of the polynomial equation is the degree of the polynomial. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Quadratic Equation:An equation of the formax2+bx+c=0ax2+bx+c=0is called a quadratic equation.a,b,care real numbers andaâ0a,b,care real numbers andaâ0 | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Zero Product Property:Ifa·b=0,a·b=0,then eithera=0a=0orb=0b=0or both. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
How to use the Zero Product PropertyStep 1.Set each factor equal to zero.Step 2.Solve the linear equations.Step 3.Check. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 1.Set each factor equal to zero. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 2.Solve the linear equations. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 3.Check. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
How to solve a quadratic equation by factoring.Step 1.Write the quadratic equation in standard form,ax2+bx+c=0.ax2+bx+c=0.Step 2.Factor the quadratic expression.Step 3.Use the Zero Product Property.Step 4.Solve the linear equations.Step 5.Check. Substitute each solution separately into the original equation. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 1.Write the quadratic equation in standard form,ax2+bx+c=0.ax2+bx+c=0. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 2.Factor the quadratic expression. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 3.Use the Zero Product Property. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 4.Solve the linear equations. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 5.Check. Substitute each solution separately into the original equation. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Zero of a Function:For any functionf, iff(x)=0,f(x)=0,thenxis a zero of the function. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
How to use a problem solving strategy to solve word problems.Step 1.Readthe problem. Make sure all the words and ideas are understood.Step 2.Identifywhat we are looking for.Step 3.Namewhat we are looking for. Choose a variable to represent that quantity.Step 4.Translateinto an equation. It may be helpful to restate the... | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 1.Readthe problem. Make sure all the words and ideas are understood. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 2.Identifywhat we are looking for. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 3.Namewhat we are looking for. Choose a variable to represent that quantity. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 4.Translateinto an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebraic equation. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 5.Solvethe equation using appropriate algebra techniques. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 6.Checkthe answer in the problem and make sure it makes sense. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
Step 7.Answerthe question with a complete sentence. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-concepts |
degree of the polynomial equation : The degree of the polynomial equation is the degree of the polynomial. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-terms |
factoring : Splitting a product into factors is called factoring. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-terms |
greatest common factor : The greatest common factor (GCF) of two or more expressions is the largest expression that is a factor of all the expressions. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-terms |
polynomial equation : A polynomial equation is an equation that contains a polynomial expression. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-terms |
quadratic equation : Polynomial equations of degree two are called quadratic equations. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-terms |
zero of the function : A value ofxxwhere the function is 0, is called a zero of the function. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-terms |
Zero Product Property : The Zero Product Property says that if the product of two quantities is zero, then at least one of the quantities is zero. | https://openstax.org/books/intermediate-algebra-2e/pages/6-key-terms |
Determine the values for which a rational expression is undefined.Step 1.Set the denominator equal to zero.Step 2.Solve the equation. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Step 1.Set the denominator equal to zero. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Step 2.Solve the equation. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Equivalent Fractions PropertyIfa,b, andcare numbers wherebâ0,câ0,bâ0,câ0,thenab=a·cb·cab=a·cb·canda·cb·c=ab.a·cb·c=ab. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
How to simplify a rational expression.Step 1.Factor the numerator and denominator completely.Step 2.Simplify by dividing out common factors. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Step 1.Factor the numerator and denominator completely. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Step 2.Simplify by dividing out common factors. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Opposites in a Rational Expressionââââ
The opposite ofaâbaâbisbâa.bâa.ââââaâbbâa=â1aâbaâbbâa=â1aâbââââAn expression and its opposite divide toâ1.â1. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Multiplication of Rational ExpressionsIfp,q,r, andsare polynomials whereqâ0,sâ0,qâ0,sâ0,thenpq·rs=prqspq·rs=prqs | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
How to multiply rational expressions.Step 1.Factor each numerator and denominator completely.Step 2.Multiply the numerators and denominators.Step 3.Simplify by dividing out common factors. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Step 1.Factor each numerator and denominator completely. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Step 2.Multiply the numerators and denominators. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Step 3.Simplify by dividing out common factors. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Division of Rational ExpressionsIfp,q,r, andsare polynomials whereqâ0,râ0,sâ0,qâ0,râ0,sâ0,thenpq÷rs=pq·srpq÷rs=pq·sr | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
How to divide rational expressions.Step 1.Rewrite the division as the product of the first rational expression and the reciprocal of the second.Step 2.Factor the numerators and denominators completely.Step 3.Multiply the numerators and denominators together.Step 4.Simplify by dividing out common factors. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Step 1.Rewrite the division as the product of the first rational expression and the reciprocal of the second. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Step 2.Factor the numerators and denominators completely. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Step 3.Multiply the numerators and denominators together. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Step 4.Simplify by dividing out common factors. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
How to determine the domain of a rational function.Step 1.Set the denominator equal to zero.Step 2.Solve the equation.Step 3.The domain is all real numbers excluding the values found in Step 2. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Step 1.Set the denominator equal to zero. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Step 2.Solve the equation. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Step 3.The domain is all real numbers excluding the values found in Step 2. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Rational Expression Addition and SubtractionIfp,q, andrare polynomials whererâ0,râ0,thenpr+qr=p+qrpr+qr=p+qrandprâqr=pâqrprâqr=pâqr | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
How to find the least common denominator of rational expressions.Step 1.Factor each expression completely.Step 2.List the factors of each expression. Match factors vertically when possible.Step 3.Bring down the columns.Step 4.Write the LCD as the product of the factors. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Step 1.Factor each expression completely. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Step 2.List the factors of each expression. Match factors vertically when possible. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Step 3.Bring down the columns. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Step 4.Write the LCD as the product of the factors. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
How to add or subtract rational expressions.Step 1.Determine if the expressions have a common denominator.Yes â go to step 2.No â Rewrite each rational expression with the LCD.Find the LCD.Rewrite each rational expression as an equivalent rational expression with the LCD.Step 2.Add or subtract the rational expressi... | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Step 1.Determine if the expressions have a common denominator.Yes â go to step 2.No â Rewrite each rational expression with the LCD.Find the LCD.Rewrite each rational expression as an equivalent rational expression with the LCD. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Yes â go to step 2. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
No â Rewrite each rational expression with the LCD.Find the LCD.Rewrite each rational expression as an equivalent rational expression with the LCD. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Find the LCD. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Rewrite each rational expression as an equivalent rational expression with the LCD. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Step 2.Add or subtract the rational expressions. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Step 3.Simplify, if possible. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
How to simplify a complex rational expression by writing it as division.Step 1.Simplify the numerator and denominator.Step 2.Rewrite the complex rational expression as a division problem.Step 3.Divide the expressions. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Step 1.Simplify the numerator and denominator. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Step 2.Rewrite the complex rational expression as a division problem. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Step 3.Divide the expressions. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
How to simplify a complex rational expression by using the LCD.Step 1.Find the LCD of all fractions in the complex rational expression.Step 2.Multiply the numerator and denominator by the LCD.Step 3.Simplify the expression. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Step 1.Find the LCD of all fractions in the complex rational expression. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Step 2.Multiply the numerator and denominator by the LCD. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Step 3.Simplify the expression. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
How to solve equations with rational expressions.Step 1.Note any value of the variable that would make any denominator zero.Step 2.Find the least common denominator of all denominators in the equation.Step 3.Clear the fractions by multiplying both sides of the equation by the LCD.Step 4.Solve the resulting equation.Ste... | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Step 1.Note any value of the variable that would make any denominator zero. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Step 2.Find the least common denominator of all denominators in the equation. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Step 3.Clear the fractions by multiplying both sides of the equation by the LCD. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Step 4.Solve the resulting equation. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Step 5.Check:If any values found in Step 1 are algebraic solutions, discard them.Check any remaining solutions in the original equation. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
If any values found in Step 1 are algebraic solutions, discard them. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Check any remaining solutions in the original equation. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
A proportion is an equation of the formab=cd,ab=cd,wherebâ0,dâ0.bâ0,dâ0.The proportion is read âais tobascis tod.â | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Property of Similar TrianglesIfÎABCÎABCis similar toÎXYZ,ÎXYZ,then their corresponding angle measure are equal and their corresponding sides have the same ratio. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Direct VariationFor any two variablesxandy,yvaries directly withxify=kx,y=kx,wherekâ0.kâ0.The constantkis called the constant of variation.How to solve direct variation problems.Step 1.Write the formula for direct variation.Step 2.Substitute the given values for the variables.Step 3.Solve for the constant of variatio... | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
For any two variablesxandy,yvaries directly withxify=kx,y=kx,wherekâ0.kâ0.The constantkis called the constant of variation. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
How to solve direct variation problems.Step 1.Write the formula for direct variation.Step 2.Substitute the given values for the variables.Step 3.Solve for the constant of variation.Step 4.Write the equation that relatesxxandy.y. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Step 1.Write the formula for direct variation. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Step 2.Substitute the given values for the variables. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Step 3.Solve for the constant of variation. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Step 4.Write the equation that relatesxxandy.y. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Inverse VariationFor any two variablesxandy,yvaries inversely withxify=kx,y=kx,wherekâ0.kâ0.The constantkis called the constant of variation.How to solve inverse variation problems.Step 1.Write the formula for inverse variation.Step 2.Substitute the given values for the variables.Step 3.Solve for the constant of vari... | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
For any two variablesxandy,yvaries inversely withxify=kx,y=kx,wherekâ0.kâ0.The constantkis called the constant of variation. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
How to solve inverse variation problems.Step 1.Write the formula for inverse variation.Step 2.Substitute the given values for the variables.Step 3.Solve for the constant of variation.Step 4.Write the equation that relatesxandy. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Step 1.Write the formula for inverse variation. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Step 2.Substitute the given values for the variables. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
Step 3.Solve for the constant of variation. | https://openstax.org/books/intermediate-algebra-2e/pages/7-key-concepts |
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