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Step 5.Factor by grouping. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Step 6.Check by multiplying the factors. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Choose a strategy to factor polynomials completely (updated):Step 1.Is there a greatest common factor? Factor it.Step 2.Is the polynomial a binomial, trinomial, or are there more than three terms?If it is a binomial, right now we have no method to factor it.If it is a trinomial of the formx2+bx+cx2+bx+cUndo FOIL(x)(x)(... | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Step 1.Is there a greatest common factor? Factor it. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Step 2.Is the polynomial a binomial, trinomial, or are there more than three terms?If it is a binomial, right now we have no method to factor it.If it is a trinomial of the formx2+bx+cx2+bx+cUndo FOIL(x)(x)(x)(x).If it is a trinomial of the formax2+bx+cax2+bx+cUse Trial and Error or the âacâ method.If it has more t... | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Step 3.Check by multiplying the factors. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Factor perfect square trinomialsSeeExample 7.42.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)2S... | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Factor differences of squaresSeeExample 7.47.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 th... | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Factor sum and difference of cubesTo factor the sum or difference of cubes: SeeExample 7.54.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 ... | https://openstax.org/books/elementary-algebra-2e/pages/7-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/elementary-algebra-2e/pages/7-key-concepts |
Step 2.Write them as cubes. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Step 3.Use either the sum or difference of cubes pattern. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Step 4.Simplify inside the parentheses | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Step 5.Check by multiplying the factors. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
General Strategy for Factoring Polynomials SeeFigure 7.3. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
How to Factor PolynomialsStep 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 difference?Of squares? Factor as th... | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Step 1.Is there a greatest common factor? Factor it out. | https://openstax.org/books/elementary-algebra-2e/pages/7-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/elementary-algebra-2e/pages/7-key-concepts |
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. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Of squares? Sums of squares do not factor. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Of cubes? Use the sum of cubes pattern. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Of squares? Factor as the product of conjugates. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Of cubes? Use the difference of cubes pattern. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
If it is a trinomial:Is it of the formx2+bx+cx2+bx+c? Undo FOIL.Is it of the formax2+bx+cax2+bx+c?If âaâ and âcâ are squares, check if it fits the trinomial square pattern.Use the trial and error or âacâ method. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
If âaâ and âcâ are squares, check if it fits the trinomial square pattern. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Use the trial and error or âacâ method. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
If it has more than three terms:Use the grouping method. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Step 3.Check. Is it factored completely? Do the factors multiply back to the original polynomial? | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Zero Product PropertyIfa·b=0a·b=0, then eithera=0a=0orb=0b=0or both. SeeExample 7.69. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Solve a quadratic equation by factoringTo solve a quadratic equation by factoring: SeeExample 7.73.Step 1.Write the quadratic equation in standard form,ax2+bx+c=0ax2+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. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Step 1.Write the quadratic equation in standard form,ax2+bx+c=0ax2+bx+c=0. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Step 2.Factor the quadratic expression. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Step 3.Use the Zero Product Property. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Step 4.Solve the linear equations. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Step 5.Check. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Use a problem solving strategy to solve word problemsSeeExample 7.80.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 res... | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Step 1.Readthe problem. Make sure all the words and ideas are understood. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Step 2.Identifywhat we are looking for. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Step 3.Namewhat we are looking for. Choose a variable to represent that quantity. | https://openstax.org/books/elementary-algebra-2e/pages/7-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 algebra equation. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Step 5.Solvethe equation using good algebra techniques. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Step 6.Checkthe answer in the problem and make sure it makes sense. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Step 7.Answerthe question with a complete sentence. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
difference of squares pattern : Ifaandbare real numbers, | https://openstax.org/books/elementary-algebra-2e/pages/7-key-terms |
factoring : Factoring is splitting a product into factors; in other words, it is the reverse process of multiplying. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-terms |
greatest common factor : The greatest common factor is the largest expression that is a factor of two or more expressions is the greatest common factor (GCF). | https://openstax.org/books/elementary-algebra-2e/pages/7-key-terms |
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/elementary-algebra-2e/pages/7-key-terms |
prime polynomials : Polynomials that cannot be factored are prime polynomials. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-terms |
quadratic equations : are equations in which the variable is squared. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-terms |
sum and difference of cubes pattern : a3+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/elementary-algebra-2e/pages/7-key-terms |
Zero Product Property : The Zero Product Property states that, if the product of two quantities is zero, at least one of the quantities is zero. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-terms |
Determine the Values for Which a Rational Expression is UndefinedStep 1.Set the denominator equal to zero.Step 2.Solve the equation, if possible. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Step 1.Set the denominator equal to zero. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Step 2.Solve the equation, if possible. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Simplified Rational ExpressionA rational expression is considered simplified if there are no common factors in its numerator and denominator. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
A rational expression is considered simplified if there are no common factors in its numerator and denominator. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Simplify a Rational ExpressionStep 1.Factor the numerator and denominator completely.Step 2.Simplify by dividing out common factors. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Step 1.Factor the numerator and denominator completely. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Step 2.Simplify by dividing out common factors. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Opposites in a Rational ExpressionThe opposite ofaâbaâbisbâabâa.aâbbâa=â1aâ0,bâ0,aâbaâbbâa=â1aâ0,bâ0,aâb | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
The opposite ofaâbaâbisbâabâa.aâbbâa=â1aâ0,bâ0,aâbaâbbâa=â1aâ0,bâ0,aâb | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Multiplication of Rational ExpressionsIfp,q,r,sp,q,r,sare polynomials whereqâ0,sâ0qâ0,sâ0, thenpq·rs=prqspq·rs=prqs.To multiply rational expressions, multiply the numerators and multiply the denominators | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Ifp,q,r,sp,q,r,sare polynomials whereqâ0,sâ0qâ0,sâ0, thenpq·rs=prqspq·rs=prqs. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
To multiply rational expressions, multiply the numerators and multiply the denominators | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Multiply a Rational ExpressionStep 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/elementary-algebra-2e/pages/8-key-concepts |
Step 1.Factor each numerator and denominator completely. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Step 2.Multiply the numerators and denominators. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Step 3.Simplify by dividing out common factors. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Division of Rational ExpressionsIfp,q,r,sp,q,r,sare polynomials whereqâ0,râ0,sâ0qâ0,râ0,sâ0, thenpq÷rs=pq·srpq÷rs=pq·sr.To divide rational expressions multiply the first fraction by the reciprocal of the second. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Ifp,q,r,sp,q,r,sare polynomials whereqâ0,râ0,sâ0qâ0,râ0,sâ0, thenpq÷rs=pq·srpq÷rs=pq·sr. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
To divide rational expressions multiply the first fraction by the reciprocal of the second. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Divide Rational ExpressionsStep 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/elementary-algebra-2e/pages/8-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/elementary-algebra-2e/pages/8-key-concepts |
Step 2.Factor the numerators and denominators completely. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Step 3.Multiply the numerators and denominators together. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Step 4.Simplify by dividing out common factors. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Rational Expression AdditionIfp,q,andrp,q,andrare polynomials whererâ0râ0, thenpr+qr=p+qrpr+qr=p+qrTo add rational expressions with a common denominator, add the numerators and place the sum over the common denominator. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Ifp,q,andrp,q,andrare polynomials whererâ0râ0, thenpr+qr=p+qrpr+qr=p+qr | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
To add rational expressions with a common denominator, add the numerators and place the sum over the common denominator. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Rational Expression SubtractionIfp,q,andrp,q,andrare polynomials whererâ0râ0, thenprâqr=pâqrprâqr=pâqrTo subtract rational expressions, subtract the numerators and place the difference over the common denominator. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Ifp,q,andrp,q,andrare polynomials whererâ0râ0, thenprâqr=pâqrprâqr=pâqr | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
To subtract rational expressions, subtract the numerators and place the difference over the common denominator. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Find the Least Common Denominator of Rational ExpressionsStep 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.Multiply the factors. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Step 1.Factor each expression completely. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Step 2.List the factors of each expression. Match factors vertically when possible. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Step 3.Bring down the columns. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Step 4.Multiply the factors. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Add or Subtract Rational ExpressionsStep 1.Determine if the expressions have a common denominator.Yesâ go to step 2.Noâ 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 expressions.Step 3... | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Step 1.Determine if the expressions have a common denominator.Yesâ go to step 2.Noâ 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/elementary-algebra-2e/pages/8-key-concepts |
Find the LCD. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Rewrite each rational expression as an equivalent rational expression with the LCD. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Step 2.Add or subtract the rational expressions. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Step 3.Simplify, if possible. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
To Simplify a Rational Expression by Writing it as DivisionStep 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/elementary-algebra-2e/pages/8-key-concepts |
Step 1.Simplify the numerator and denominator. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Step 2.Rewrite the complex rational expression as a division problem. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Step 3.Divide the expressions. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
To Simplify a Complex Rational Expression by Using the LCDStep 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/elementary-algebra-2e/pages/8-key-concepts |
Step 1.Find the LCD of all fractions in the complex rational expression. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Step 2.Multiply the numerator and denominator by the LCD. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
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