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Factorization and Algebraic Fractions
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Multiplication and Division Operations in Algebraic Fractions
When we want to multiply or divide algebraic fractions, we will use the same tools that we use for the multiplication or division of common fractions with some small differences.
Steps to carry out for the multiplication of algebraic fractions 1:
Let's try to extract the common factor. This can be the variable or any free number.
How is the solution set found? We will make all the denominators we have equal to 0 and find the solution. The solution set will be X: different from what causes our denominator to equal zero.
Let's simplify the fractions with determination.
Multiply numerator by numerator and denominator by denominator as in any fraction.
Steps to carry out for the division of algebraic fractions2:
We will convert the division exercise into a multiplication one, as we do with common fractions. How will we do it correctly? We will leave the first fraction as it is, change the division sign to a multiplication sign, and invert the fraction that appears after the sign. That is, numerator instead of denominator and denominator instead of numerator.
We will act according to the rules of multiplication of algebraic fractions:
Let's try to extract the common factor. This can be the unknown or any free number.
If this is not enough, we will factorize using short multiplication formulas and with trinomials.
Let's find the solution set.
How is the solution set found? We will make all the denominators we have equal to 0 and find the solution. The solution set will be X: different from what causes our denominator to equal zero.
Let's simplify the fractions with determination.
Multiply numerator by numerator and denominator by denominator as in any fraction.
Let's look at an example of multiplying algebraic fractions
x+3x+2βΓx2β43x+9β=
Let's try to factorize by extracting the common factor and with the shortcut multiplication formulas, and we will obtain: x+3x+2βΓ(xβ2)(x+23(x+3)β=
Let's find the solution set:
xξ =β3,2,β2
Let's reduce the fractions and we will obtain:
1Γ(xβ2)3β= Multiply and it will give us: xβ23β
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Let's convert the division exercise into a multiplication one:
x2β3x+2x2β8x+15βΓx2β9xβ1β= Now, let's factor and we will get: (xβ2)(xβ1)(xβ5)(xβ3)βΓ(xβ3)(x+3)xβ1β= Let's find the solution set: xξ =2,1,3,β3
Let's simplify, we will get:
xβ2xβ5βΓx+31β
Let's multiply and it will give us: (xβ2)(x+3)xβ5β
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