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7.1: Multiply Rational Expressions - Mathematics


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Multiplying Rational Expressions – 9th Grade Algebra

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7.1: Multiply Rational Expressions - Mathematics

Equations

When faced with a rational equation, we encounter two difficulties. The first difficulty we face is that we are dealing with equations with complication expressions in denominators. Secondly, we face the possibility of what are known as extraneous solutions, that is, false solutions that appear. In this article we will deal with both difficulties.

Let’s try to solve for (a) in the following equation:

(Large frac<> <<2+ 18a + 28>> + Large frac<1> <<2+ 18a + 28>> = Large frac<1> <<+ 9a + 14>>)

We should try to multiply both sides of the equation by some expressions or numbers which would eliminate our denominators. But what? If we do some factoring, the answer will become clearer:

(Large frac<> <<2+ 18a + 28>> + Large frac<1> <<2+ 18a + 28>> = Large frac<1> <<+ 9a + 14>>)

(Large frac<> <<2left( <+ 9a + 14> ight)>> + Large frac<1> <<2left( <+ 9a + 14> ight)>> = Large frac<1> <<+ 9a + 14>>)

Hopefully you can see that all three terms in this equation contain the expression ( + 9a + 14). If we multiply both sides of the equation by that expression, we will do some major cancelling in our denominators.

(Large frac<> <<2left( <+ 9a + 14> ight)>> + Large frac<1> <<2left( <+ 9a + 14> ight)>> = Large frac<1> <<+ 9a + 14>>)

Wow! That operation really cleaned up the equation. Let’s clean up some more by multiplying both sides by 2, and then solving for (a)!. We have

(left( 2 ight)left( > <2>+ Large frac<1><2>> ight) = left( 1 ight)left( 2 ight))

Here we must be careful. When we multiply both sides of an equation by an expression containing variables, we must check for extraneous, or false, solutions. The reasoning behind this has to do with the fact that division by 0 is undefined. You can do more research to learn about extraneous solutions if you wish. But all it boils down to is that we must check the solution we got in our original equation. We have

(Large frac<<4 - 3>><<2<^2> + 18left( 4 ight) + 28>> + Large frac<1><<2<^2> + 18left( 4 ight) + 28>> = Large frac<1> <<<4^2>+ 9left( 4 ight) + 14>>)

The solution checks. Therefore (a = 4) is a solution to our original equation. Let’s look at another example. We want to solve for (p) in

First multiply both sides by (p - 5)

Again, check for extraneous solutions

(Large frac<8> <7>= 1 + Large frac<1><7>)

The solution checks. So (p = 12) is a solution to the original equation.

Below you can download some free math worksheets and practice.


How to Multiply & Divide Rational Expressions


Note that we keep track of the $x$-values that would cause the calculation to be undefined at any step.

(Optional) Some instructors will ask you to expand the numerator and denominators when possible.

Note: The answer should include the restrictions we found in Step 3.

Problem 2

Factor the numerators and denominators.

Write the product as a single fraction.

Problem 3

Factor the numerators and the denominators.

Rewrite the product as a single fraction.

Problem 4

Factor the numerators and denominators.

Rewrite the product as a single fraction.

Problem 5

Factor the numerators and denominators.

Note that our final answer will need to restrict the values of $x$ so that $x eq -5, 0$.

Rewrite the division as a product with the reciprocal of the second expression.

Note that we now also need to make sure our answer restricts $x$ so $x eq -2, 2$.

Rewrite the product as a single fraction.

Note that the other restrictions we identified ($x eq -2$ and $x eq 0$) are still implied in the expression itself.


To multiply rational expressions, we apply the steps below:

  • Completely factor out denominators and numerators of both fractions.
  • Cancel out common terms in the numerator and denominator.
  • Now rewrite the remaining terms both in the numerator and denominator.

Use the algebraic identities below to help you in factoring the polynomials:

  • (a² – b²) = (a + b) (a – b)
  • (x² – 4²) = (x + 4) (x – 4)
  • (x² – 2²) = (x + 2) (x – 2)
  • (a³ + b³) = (a + b) (a² – a b + b²)

Simplify (x² – 2x) / (x + 2) * (3 x + 6)/ (x – 2)

Cancel out common terms in numerators and denominators of both fractions to get

Solve [(x 2 – 3x – 4)/ (x 2 – x -2)] * [(x 2 – 4)/ (x 2 -+ x -20)]

First, factor the numerators and denominators of both fractions.

[(x – 4) (x + 1)/ (x + 1) (x – 2)] * [(x + 2) (x – 2)/ (x – 4) (x + 5)]

Cancel out common terms and rewrite the remaining terms

Multiply [(12x – 4x 2 )/ (x 2 + x – 12)] * [(x 2 + 2x – 8)/x 3 – 4x)]

Factor the rational expressions.

⟹ [-4x (x – 3)/ (x – 3) (x + 4)] * [(x – 2) (x + 4)/x (x + 2) (x – 2)]

Reduce the fractions by cancelling common terms in the numerators and denominators to get

Multiply [(2x 2 + x – 6)/ (3x 2 – 8x – 3)] * [(x 2 – 7x + 12)/ (2x 2 – 7x – 4)]

⟹ [(2x – 3) (x + 2)/ (3x + 1) (x – 3)] * [(x – 30(x – 4)/ (2x + 1) (x – 4)]

Cancel out common terms in the numerators and denominators and rewrite the remaining terms.

Simplify [(x² – 81)/ (x² – 4)] * [(x² + 6 x + 8)/ (x² – 5 x – 36)]

Factor the numerators and denominators of each fraction.

⟹ [(x + 9) (x – 9)/ (x + 2) (x – 2)] * [(x + 2) (x + 4)/ (x – 9) (x + 4)]

On cancelling common terms, we get

Simplify [(x² – 3 x – 10)/ (x² – x – 20)] * [(x² – 2 x + 4)/ (x³ + 8)]

Factor out (x³ + 8) using the algebraic identity (a³ + b³) = (a + b) (a² – a b + b²).

[(x² – 3 x – 10)/ (x² – x – 20)] * [(x² – 2 x + 4)/ (x³ + 8)] = [(x – 5) (x + 2)/ (x – 5) (x + 4)] * [(x² – 2 x + 4)/ (x + 2) (x² – 2 x + 4)]

Now, cancel out common terms to get

Simplify [(x + 7)/ (x² + 14 x + 49)] * [(x² + 8x + 7)/ (x + 1)]

= [(x + 7)/ (x + 7) (x + 7)] * [(x + 1) (x + 7)/ (x + 1)]

On cancelling common terms, we get the answer as

Use the algebraic identity (a² – b²) = (a + b) (a – b) to factor (x² – 16) and (x² – 4).

Also apply the identity (a³ + b³) = (a + b) (a² – a b + b²) to factor (x³ + 64).

= [(x + 4) (x – 4)/)/ (x – 2)] * [(x + 2) (x – 2)/ (x² – 4x + 16)]

Cancel common terms to get

Simplify [(x² – 9 y²)/ (3 x – 3y)] * [(x² – y²)/ (x² + 4 x y + 3 y²)]

Apply the algebraic identity (a²-b²) = (a + b) (a – b) to factor (x²- (3y) ² and (x² – y²)


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Multiplying Rational Expressions



A rational expression is a fraction in which either the numerator, or the denominator, or both the numerator and the denominator are algebraic expressions.

When two fractions are multiplied, we multiply the numerators of the fractions to form the new numerator and we do the same for the denominators. This is the same with rational expressions. If there are common factors in both numerator and denominator of the two rational expressions then we may cancel them before we multiply.

Simplify the following expressions:

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.


Perfect Mathematics

In the algebraic expression the variable does not occur in the fraction or negative index. While using the rational expression calculator the variable that occur in the fractional notation. The calculator show the error mistake. while in the multiply algebraic expression the calculator will be mention in the integer it may not occur any fraction or negative variable.

For example 5x2- 3x +2 this is the algebraic expression
An rational expression of the form A(x) * B(x) where A(x) and B(x) are two polynomials over the set of real numbers and QA(x) ? 0 is called a rational expression.
For example 2/x^2 , ((x^4+x^3+x+1))/((x+5)), are rational expressions.

Rational Expression on Multiplying Rational Expressions Calculator

Problem in rational expression calculator In this expression the variable only in the integer not in the fraction form.
1. Simplify: (x2-x-6)/(x2+5x+6)
= ((x^2-x-6))/((x^2+5x+6))
= ((x-3)(x+2))/((x+2)(x+3))
= ((x-3))/((x+3))
Multiplication of rational expressions
The product of rational expression in the form. The resulting expression is then reduced to its lowest form. If p(x)*g(x)
= (p(x))/(q(x)) + (g(x))/(h(x))
= (g(x))/(h(x)) * (g(x))/(h(x))
In this expression the multiplication in the rational expression are in the status of variable must be in the integers not in the fraction. The multiplication rational expression calculator is reduced to its lowest form.


Multiplying Algebraic Expressions

In these lessons, we will learn how to multiply algebraic expressions.

The following diagram shows some expansions, that are useful to remember, when multiplying two algebraic expressions or binomials. Scroll down the page for more examples and solutions on how to expand expressions.

How to Multiply a Term and an Algebraic Expression?

We will first consider examples of multiplying a term and an algebraic expression.

How to Multiply Two Algebraic Expressions?

Next, we will also consider the multiplication of two algebraic expressions: (a + b)(c + d)

Such an operation is called &lsquoexpanding the expression &rsquo.
To expand the expression, we multiply each term in the first pair of brackets by every term in the second pair of brackets.

b) (a + b) 2
= (a + b)(a + b) = a(a + b) + b(a + b)
= a 2 + ab + ab + b 2
= a 2 + 2ab + b 2

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.


How To : Multiply rational expressions with opposite signs

In this video the instructor shows how to multiply and write rational expressions in lowest terms. The fist thing you need to do is cancel out the common factors in the numerator and the denominator. You can cancel a term in the top with a term in the bottom even if they are diagonal as long as one is in numerator and the other is in the denominator. After cancellation if you have a term in numerator and an identical term in the denominator but with opposite signs, then pull out the negative sign from one of the terms which makes both the terms in numerator and denominator same which can be now canceled out and the rest of the equation simplified. This video shows how to simplify rational expressions when terms in numerator and denominator differ by signs.

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Watch the video: Multiply Rational Expressions (December 2021).