Problem 45
Question
Multiply. $$\frac{2 x^{2}-5 x}{2 x y+y} \cdot \frac{2 x y^{2}+y^{2}}{5 x^{2}-2 x^{3}}$$
Step-by-Step Solution
Verified Answer
The result, after multiplication and simplification of the given fractional expressions, is \( \frac{{2x^2y^2 -5xy^2}}{{y(5x^2-2x^3)}} \)
1Step 1: Multiplication of Numerators and Denominators
First, multiply the numerators together and then the denominators, respectively. This will result in the following: \( \frac{{(2x^2 - 5x)(2xy^2 + y^2)}}{{(2xy + y)(5x^2 - 2x^3)}} \)
2Step 2: Simplify
Next, distribute and combine the like terms for both the numerator and denominator. The numerator becomes \( 4x^3y^2+2x^2y^2 -10xy^3-5xy^2 \) whilst the denominator becomes \( 10x^3y+5x^2y-4x^4y-2x^3y \)
3Step 3: Further Simplification
We can further simplify the expressions by factoring out the common terms both in the numerator and denominator. This gives \( \frac{{2x^2y^2(2x + y) -5xy^2(2x+y)}}{{y(2x+y)(5x^2-2x^3)}} \)
4Step 4: Final Simplification
The final step involves canceling out any common factors in both the numerator and denominator. Here, we shall cancel the common factor \( (2x+y) \). Our final simplified expression is \( \frac{{2x^2y^2 -5xy^2}}{{y(5x^2-2x^3)}} \)
Key Concepts
Multiplication of PolynomialsFactoring Algebraic ExpressionsCancellation in Algebraic Fractions
Multiplication of Polynomials
When multiplying polynomials, one must understand the distributive property, which allows us to multiply each term in the first polynomial by every term in the second polynomial. Let's consider the given algebraic fractions' numerators.
Seen in our exercise, we have two polynomials in the numerators: \(2x^2 - 5x)\) and \(2xy^2 + y^2)\). To multiply these, we apply the distributive property, also known as the FOIL method (First, Outside, Inside, Last) when dealing with binomials, though this method extends to any number of terms. It entails multiplying each term of the first polynomial by each term of the second polynomial and combining like terms. Here's how it looks step by step:
Seen in our exercise, we have two polynomials in the numerators: \(2x^2 - 5x)\) and \(2xy^2 + y^2)\). To multiply these, we apply the distributive property, also known as the FOIL method (First, Outside, Inside, Last) when dealing with binomials, though this method extends to any number of terms. It entails multiplying each term of the first polynomial by each term of the second polynomial and combining like terms. Here's how it looks step by step:
- Multiply \(2x^2\) by \(2xy^2\), which gives \(4x^3y^2\).
- Multiply \(2x^2\) by \(y^2\), resulting in \(2x^2y^2\).
- Then, multiply \( -5x\) by \(2xy^2\), giving \( -10xy^3\).
- Lastly, multiply \( -5x\) by \(y^2\), which is \( -5xy^2\).
Factoring Algebraic Expressions
Factoring is the process of breaking down an expression into simpler components called 'factors' that, when multiplied together, give the original expression. It is an integral part of simplifying algebraic fractions.
In our example, once we have the expanded form of the numerator and denominator post multiplication, we look for common factors that can be factored out. This is done to simplify the expression before performing any cancellation. Factoring out common terms in the numerator and denominator makes the expression neater and often, simpler to work with. For instance, \(2x^2y^2(2x + y) - 5xy^2(2x + y)\) shows that the term \(2x + y)\) is a common factor in both terms of the numerator. Similarly, in the denominator, we note that the common factor \(y\) can be factored out of the terms. Factoring not only helps with simplification but also prepares the expression for the final step of cancellation. It is important to always be on the lookout for such factors to make simplification possible.
In our example, once we have the expanded form of the numerator and denominator post multiplication, we look for common factors that can be factored out. This is done to simplify the expression before performing any cancellation. Factoring out common terms in the numerator and denominator makes the expression neater and often, simpler to work with. For instance, \(2x^2y^2(2x + y) - 5xy^2(2x + y)\) shows that the term \(2x + y)\) is a common factor in both terms of the numerator. Similarly, in the denominator, we note that the common factor \(y\) can be factored out of the terms. Factoring not only helps with simplification but also prepares the expression for the final step of cancellation. It is important to always be on the lookout for such factors to make simplification possible.
Cancellation in Algebraic Fractions
Cancellation is the process of reducing algebraic fractions to their simplest form by eliminating common factors from the numerator and denominator. Imagine it like 'crossing out' the same number from the top and bottom of a fraction.
In the given problem, once we have factored the polynomials, we can perform cancellation. We observe that the term \(2x + y)\) is common in both the numerator and denominator, which means we can cancel it out. This process is a simplifying step that should be done carefully to avoid canceling out terms that are not common factors. An incorrect cancellation can lead to a wrong answer. After the correct cancellation of the common factor \(2x + y)\), we are left with a simpler fraction \(\frac{{2x^2y^2 - 5xy^2}}{{y(5x^2 - 2x^3)}}\).Remember, we can only cancel terms that are factors, which means they must be multiplied by the rest of the expression. We cannot cancel terms that are added or subtracted. Cancellation helps in reducing the expression to its lowest terms, which is particularly useful in solving and understanding algebraic fractions.
In the given problem, once we have factored the polynomials, we can perform cancellation. We observe that the term \(2x + y)\) is common in both the numerator and denominator, which means we can cancel it out. This process is a simplifying step that should be done carefully to avoid canceling out terms that are not common factors. An incorrect cancellation can lead to a wrong answer. After the correct cancellation of the common factor \(2x + y)\), we are left with a simpler fraction \(\frac{{2x^2y^2 - 5xy^2}}{{y(5x^2 - 2x^3)}}\).Remember, we can only cancel terms that are factors, which means they must be multiplied by the rest of the expression. We cannot cancel terms that are added or subtracted. Cancellation helps in reducing the expression to its lowest terms, which is particularly useful in solving and understanding algebraic fractions.
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