MTH120 College Algebra Chapter 1.5

 1.5 Factoring Polynomials:

Factoring the greatest common factor (GCF) of a polynomial involves finding the largest expression that can be factored out of all the terms in the polynomial. Factoring the GCF simplifies the polynomial and makes it easier to work with. Here's how you can factor the GCF of a polynomial:

1. Identify the Polynomial: Start by identifying the polynomial you want to factor. For example, let's say you have the polynomial: $6{�}^2+9�$.

2. Identify the Terms: Break down the polynomial into its individual terms. In the example above, there are two terms: $6{�}^2$ and $9�$.

3. Find the GCF of the Coefficients: Look at the coefficients (the numbers in front of the variables) of all the terms. In this case, the coefficients are 6 and 9. Find the greatest common factor of these coefficients. In this example, the GCF of 6 and 9 is 3.

4. Write the GCF: Write down the GCF you found in the previous step. In this case, it's 3.

5. Factor Out the GCF: Divide each term in the polynomial by the GCF and write the result inside parentheses. This is effectively factoring out the GCF: $3(2{�}^2+3�)$.

6. Simplify: If possible, further simplify the polynomial inside the parentheses. In this case, you can see that both terms inside the parentheses have a common factor of $�$. Factor that out: $3�(2�+3)$.

Now, you have factored the GCF of the polynomial $6{�}^2+9�$, and the result is $3�(2�+3)$. This is the fully factored form of the original polynomial.

Factoring the GCF is a useful first step when factoring more complex polynomials, as it often makes the polynomial easier to work with and can lead to further factorization if applicable.

Let's go through a couple of examples of factoring the greatest common factor (GCF) from polynomial expressions step by step.

Example 1: Factor the GCF from the polynomial $12{�}^3+18{�}^2$.

Step 1: Identify the Polynomial: The given polynomial is $12{�}^3+18{�}^2$.

Step 2: Identify the Terms: There are two terms in this polynomial: $12{�}^3$ and $18{�}^2$.

Step 3: Find the GCF of the Coefficients: The coefficients of the terms are 12 and 18. The GCF of 12 and 18 is 6.

Step 4: Write the GCF: Write down the GCF found in the previous step, which is 6.

Step 5: Factor Out the GCF: Divide each term in the polynomial by the GCF and write the result inside parentheses: $6{�}^2(2�+3)$.

Step 6: Simplify: In this case, no further simplification is possible. So, the fully factored form of the polynomial $12{�}^3+18{�}^2$ is $6{�}^2(2�+3)$.

Example 2: Factor the GCF from the polynomial $15{�}^2�-9�{�}^2$.

Step 1: Identify the Polynomial: The given polynomial is $15{�}^2�-9�{�}^2$.

Step 2: Identify the Terms: There are two terms in this polynomial: $15{�}^2�$ and $-9�{�}^2$.

Step 3: Find the GCF of the Coefficients: The coefficients of the terms are 15 and -9. The GCF of 15 and -9 is 3.

Step 4: Write the GCF: Write down the GCF found in the previous step, which is 3.

Step 5: Factor Out the GCF: Divide each term in the polynomial by the GCF and write the result inside parentheses: $3��(5�-3�)$.

Step 6: Simplify: In this case, no further simplification is possible. So, the fully factored form of the polynomial $15{�}^2�-9�{�}^2$ is $3��(5�-3�)$.

These examples demonstrate how to factor out the greatest common factor (GCF) from polynomial expressions step by step. The key is to identify the terms, find the GCF of their coefficients, and then factor it out.

Factoring the greatest common factor (GCF) from a polynomial expression involves finding the largest expression that can be factored out of all the terms in the polynomial. This process simplifies the polynomial and makes it easier to work with. Let's go through an example step by step:

Example: Factor the GCF from the polynomial $24{�}^3-36{�}^2+60�$.

Step 1: Identify the Polynomial: The given polynomial is $24{�}^3-36{�}^2+60�$.

Step 2: Identify the Terms: There are three terms in this polynomial: $24{�}^3$, $-36{�}^2$, and $60�$.

Step 3: Find the GCF of the Coefficients: Look at the coefficients (the numbers in front of the variables) of all the terms. In this case, the coefficients are 24, -36, and 60. Find the greatest common factor of these coefficients. The GCF of 24, -36, and 60 is 12.

Step 4: Write the GCF: Write down the GCF you found in the previous step, which is 12.

Step 5: Factor Out the GCF: Divide each term in the polynomial by the GCF (12) and write the result inside parentheses: $12(2{�}^3-3{�}^2+5�)$.

Step 6: Simplify (if possible): In this case, no further simplification is possible for the expression inside the parentheses. So, the fully factored form of the polynomial $24{�}^3-36{�}^2+60�$ is $12(2{�}^3-3{�}^2+5�)$.

This is the polynomial with the greatest common factor factored out.

The key steps are to identify the terms, find the GCF of their coefficients, and then factor out the GCF by dividing each term by it. This process simplifies the polynomial and often makes it easier to further factor or perform other operations.

Factoring a trinomial with a leading coefficient of 1 is a common algebraic task. A trinomial is a polynomial with three terms. When the leading coefficient (the coefficient of the highest-degree term) is 1, the factoring process is simplified because you only need to find two numbers that multiply to the constant term (the last term) and add up to the coefficient of the linear term (the middle term). Here's how to factor a trinomial with a leading coefficient of 1:

Example: Factor the trinomial ${�}^2+5�+6$.

Step 1: Identify the Trinomial: The given trinomial is ${�}^2+5�+6$.

Step 2: Find Two Numbers: We need to find two numbers that multiply to the constant term (6) and add up to the coefficient of the linear term (5). In this case, the two numbers are 2 and 3 because $2\cdot 3=6$ and $2+3=5$.

Step 3: Rewrite the Middle Term: Rewrite the middle term ($5�$) using the two numbers found in the previous step. In this case, we rewrite $5�$ as $2�+3�$: ${�}^2+2�+3�+6$

Step 4: Group and Factor by Grouping: Now, group the terms in pairs and factor by grouping: $({�}^2+2�)+(3�+6)$

Factor each group separately: $�(�+2)+3(�+2)$

Step 5: Factor Out the Common Factor: Notice that both groups have a common factor of $(�+2)$. Factor it out: $(�+2)(�+3)$

So, the fully factored form of the trinomial ${�}^2+5�+6$ is $(�+2)(�+3)$.

This is the factored expression of the trinomial, which represents the polynomial as a product of two binomials.

The key in this process is to find two numbers that satisfy the conditions mentioned in Step 2 and then use these numbers to rewrite the middle term, allowing you to factor by grouping and simplify the expression.

Factoring a trinomial in the form ${�}^2+��+�$ involves finding two numbers that multiply to the constant term ($�$) and add up to the coefficient of the linear term ($�$). Once you have these two numbers, you can rewrite the trinomial and then factor it. Here's the step-by-step process:

Example: Factor the trinomial ${�}^2+7�+12$.

Step 1: Identify the Trinomial: The given trinomial is ${�}^2+7�+12$.

Step 2: Find Two Numbers: You need to find two numbers that multiply to the constant term (12) and add up to the coefficient of the linear term (7). In this case, the two numbers are 3 and 4 because $3\cdot 4=12$ and $3+4=7$.

Step 3: Rewrite the Middle Term: Rewrite the middle term ($7�$) using the two numbers found in the previous step. In this case, rewrite $7�$ as $3�+4�$: ${�}^2+3�+4�+12$

Step 4: Group and Factor by Grouping: Now, group the terms in pairs and factor by grouping: $({�}^2+3�)+(4�+12)$

Factor each group separately: $�(�+3)+4(�+3)$

Step 5: Factor Out the Common Factor: Notice that both groups have a common factor of $(�+3)$. Factor it out: $(�+3)(�+4)$

So, the fully factored form of the trinomial ${�}^2+7�+12$ is $(�+3)(�+4)$.

This is the factored expression of the trinomial ${�}^2+7�+12$, represented as a product of two binomials.

The key is to find two numbers that multiply to the constant term ($�$) and add up to the coefficient of the linear term ($�$), which allows you to rewrite and factor the trinomial.

Factoring a trinomial with a leading coefficient of 1 is a common algebraic task. When the leading coefficient (the coefficient of the highest-degree term) is 1, the factoring process is simplified because you only need to find two numbers that multiply to the constant term (the last term) and add up to the coefficient of the linear term (the middle term). Here's how to factor a trinomial with a leading coefficient of 1:

Example: Factor the trinomial ${�}^2+5�+6$.

Step 1: Identify the Trinomial: The given trinomial is ${�}^2+5�+6$.

Step 2: Find Two Numbers: You need to find two numbers that multiply to the constant term (6) and add up to the coefficient of the linear term (5). In this case, the two numbers are 2 and 3 because $2\cdot 3=6$ and $2+3=5$.

Step 3: Rewrite the Middle Term: Rewrite the middle term ($5�$) using the two numbers found in the previous step. In this case, we rewrite $5�$ as $2�+3�$: ${�}^2+2�+3�+6$

Step 4: Group and Factor by Grouping: Now, group the terms in pairs and factor by grouping: $({�}^2+2�)+(3�+6)$

Factor each group separately: $�(�+2)+3(�+2)$

Step 5: Factor Out the Common Factor: Notice that both groups have a common factor of $(�+2)$. Factor it out: $(�+2)(�+3)$

So, the fully factored form of the trinomial ${�}^2+5�+6$ is $(�+2)(�+3)$.

This is the factored expression of the trinomial, represented as a product of two binomials.

The key in this process is to find two numbers that satisfy the conditions mentioned in Step 2 and then use these numbers to rewrite the middle term, allowing you to factor by grouping and simplify the expression.

Factoring by grouping is a technique used to factor polynomials that have four or more terms by grouping terms with common factors together and then factoring out the greatest common factor (GCF) from each group. Here's a step-by-step guide on how to factor by grouping:

Step 1: Identify the Polynomial Start with the polynomial expression you want to factor. Let's use an example: $2{�}^3+4{�}^2-3�-6$.

Step 2: Group Terms Group the terms in pairs, usually the first two terms and the last two terms: $(2{�}^3+4{�}^2)-(3�+6)$.

Step 3: Factor Out the GCF from Each Group Factor out the GCF from each group separately. In the first group, the GCF is $2{�}^2$, and in the second group, the GCF is $3$. Factor them out as follows: $2{�}^2(�+2)-3(�+2)$.

Step 4: Observe the Common Factor Observe that both groups now have a common factor of $(�+2)$.

Step 5: Factor Out the Common Factor Factor out the common factor $(�+2)$ from both groups: $(�+2)(2{�}^2-3)$.

Step 6: Simplify (if possible) If you can further factor any of the remaining expressions inside the parentheses, do so. In this case, the expression $2{�}^2-3$ cannot be factored further over the real numbers, so the factored form is: $(�+2)(2{�}^2-3)$.

That's it! You've successfully factored the polynomial by grouping.

Remember that the key to factoring by grouping is to identify pairs of terms with common factors, factor out those common factors, and then look for a common factor that can be factored out from the grouped terms. This technique is particularly useful for polynomials with four or more terms.

Factoring a trinomial in the form $�{�}^2+��+�$ by grouping involves breaking it into two groups of terms, then factoring out the greatest common factor (GCF) from each group separately. Here's how to do it step by step:

Example: Factor the trinomial $3{�}^2-5�-2$.

Step 1: Identify the Trinomial: The given trinomial is $3{�}^2-5�-2$.

Step 2: Group Terms: Group the terms in pairs, usually the first two terms and the last term: $(3{�}^2-5�)-2$.

Step 3: Factor Out the GCF from Each Group: Factor out the GCF from each group separately. In the first group, the GCF is $�$, and in the second group, there is no common factor to factor out: $�(3�-5)-2$.

Step 4: Observe the Common Factor: Observe that there is no common factor between the two groups at this point.

Step 5: Factor Out the Common Factor: Factor out the common factor $(1)$ from both groups: $1(�(3�-5)-2)$.

Step 6: Simplify (if possible): Now, you can simplify the expression inside the parentheses if possible. In this case, there's no further simplification: $�(3�-5)-2$.

So, the factored form of the trinomial $3{�}^2-5�-2$ by grouping is $�(3�-5)-2$.

That's how you can factor a trinomial in the form $�{�}^2+��+�$ by grouping. Remember to look for common factors within each group of terms before attempting to factor out the greatest common factor.

Factoring a trinomial in the form $�{�}^2+��+�$ by grouping involves breaking it into two groups of terms, then factoring out the greatest common factor (GCF) from each group separately. Here's how to do it step by step:

Example: Factor the trinomial $3{�}^2-5�-2$.

Step 1: Identify the Trinomial: The given trinomial is $3{�}^2-5�-2$.

Step 2: Group Terms: Group the terms in pairs, usually the first two terms and the last term: $(3{�}^2-5�)-2$.

Step 3: Factor Out the GCF from Each Group: Factor out the GCF from each group separately. In the first group, the GCF is $�$, and in the second group, there is no common factor to factor out: $�(3�-5)-2$.

Step 4: Observe the Common Factor: Observe that there is no common factor between the two groups at this point.

Step 5: Factor Out the Common Factor: Factor out the common factor $(1)$ from both groups: $1(�(3�-5)-2)$.

Step 6: Simplify (if possible): Now, you can simplify the expression inside the parentheses if possible. In this case, there's no further simplification: $�(3�-5)-2$.

So, the factored form of the trinomial $3{�}^2-5�-2$ by grouping is $�(3�-5)-2$.

That's how you can factor a trinomial in the form $�{�}^2+��+�$ by grouping. Remember to look for common factors within each group of terms before attempting to factor out the greatest common factor.

Factoring a perfect square trinomial involves recognizing the pattern and factoring it accordingly. A perfect square trinomial is of the form ${�}^2+2��+{�}^2$ or $(�+�{)}^2$, where $�$ and $�$ are variables or constants. Here's how to factor a perfect square trinomial:

Example: Factor the perfect square trinomial ${�}^2+6�+9$.

Step 1: Identify the Trinomial: The given trinomial is ${�}^2+6�+9$.

Step 2: Recognize the Pattern: Recognize that the trinomial is a perfect square trinomial because it can be written in the form $(�+3{)}^2$.

Step 3: Factor the Perfect Square Trinomial: Factor the perfect square trinomial using the square of a binomial pattern $(�+�{)}^2={�}^2+2��+{�}^2$: $(�+3{)}^2$.

So, the factored form of the perfect square trinomial ${�}^2+6�+9$ is $(�+3{)}^2$.

In summary, factoring a perfect square trinomial involves recognizing the pattern and directly factoring it as the square of a binomial. This pattern is very common and is used for simplifying expressions in algebra.

To factor a perfect square trinomial into the square of a binomial, you need to recognize the pattern and then apply it. A perfect square trinomial is of the form ${�}^2+2��+{�}^2$ or $(�+�{)}^2$, where $�$ and $�$ can be variables or constants. Here's how to factor it:

Step 1: Identify the Perfect Square Trinomial Start with the perfect square trinomial you want to factor. For example, let's use the trinomial ${�}^2+4�+4$.

Step 2: Recognize the Pattern Check if the trinomial can be written in the form $(�+�{)}^2$. In this case, the trinomial ${�}^2+4�+4$ can be recognized as $(�+2{)}^2$.

Step 3: Factor the Perfect Square Trinomial Factor the perfect square trinomial using the square of a binomial pattern $(�+�{)}^2={�}^2+2��+{�}^2$. In our example, you have: $(�+2{)}^2$.

So, the factored form of the perfect square trinomial ${�}^2+4�+4$ is $(�+2{)}^2$.

That's how you factor a perfect square trinomial into the square of a binomial. Recognize the pattern and apply it to simplify expressions or equations.

Factoring a perfect square trinomial involves recognizing the pattern and using the formula for a perfect square trinomial. A perfect square trinomial is in the form ${�}^2+2��+{�}^2$ or $(�+�{)}^2$, where $�$ and $�$ are constants or variables. Here's how to factor a perfect square trinomial:

Step 1: Identify the Trinomial Start with the perfect square trinomial you want to factor. For example, let's use the trinomial ${�}^2+6�+9$.

Step 2: Recognize the Pattern Check if the trinomial can be expressed as the square of a binomial. In this case, the trinomial ${�}^2+6�+9$ can be recognized as $(�+3{)}^2$.

Step 3: Factor the Perfect Square Trinomial Factor the perfect square trinomial using the formula for the square of a binomial $(�+�{)}^2={�}^2+2��+{�}^2$. In our example, you have: $(�+3{)}^2$.

So, the factored form of the perfect square trinomial ${�}^2+6�+9$ is $(�+3{)}^2$.

Here's another example:

Example: Factor the perfect square trinomial $4{�}^2-12�+9$.

Step 1: Identify the Trinomial The given trinomial is $4{�}^2-12�+9$.

Step 2: Recognize the Pattern Recognize that the trinomial can be expressed as the square of a binomial. In this case, it can be written as $(2�-3{)}^2$.

Step 3: Factor the Perfect Square Trinomial Factor the perfect square trinomial using the formula for the square of a binomial $(�-�{)}^2={�}^2-2��+{�}^2$. In our example, you have: $(2�-3{)}^2$.

So, the factored form of the perfect square trinomial $4{�}^2-12�+9$ is $(2�-3{)}^2$.

Recognizing the pattern of perfect square trinomials and using the square of a binomial formula makes factoring them a straightforward process.

Factoring a difference of squares involves recognizing the pattern and using the formula for the difference of squares, which states that ${�}^2-{�}^2$ can be factored as $(�+�)(�-�)$, where $�$ and $�$ can be constants or expressions. Here's how to factor a difference of squares:

Step 1: Identify the Difference of Squares Start with the expression you want to factor and check if it can be expressed as a difference of squares. A difference of squares typically has the form ${�}^2-{�}^2$, where $�$ and $�$ are constants or expressions.

Step 2: Recognize the Pattern Recognize the expression as a difference of squares, where $�$ and $�$ are both squared terms.

Step 3: Factor the Difference of Squares Factor the difference of squares using the formula $(�+�)(�-�)$. In other words, replace ${�}^2-{�}^2$ with $(�+�)(�-�)$.

Here are some examples to illustrate this:

Example 1: Factor the difference of squares ${�}^2-25$.

Step 1: Identify the Difference of Squares The given expression is ${�}^2-25$, which is in the form of a difference of squares.

Step 2: Recognize the Pattern Recognize that ${�}^2-25$ is a difference of squares because ${�}^2$ is the square of $�$, and $25$ is the square of $5$.

Step 3: Factor the Difference of Squares Factor the difference of squares using the formula $(�+�)(�-�)$: ${�}^2-25=(�+5)(�-5)$.

So, the factored form of ${�}^2-25$ is $(�+5)(�-5)$.

Example 2: Factor the difference of squares $16{�}^2-9{�}^2$.

Step 1: Identify the Difference of Squares The given expression is $16{�}^2-9{�}^2$, which is in the form of a difference of squares.

Step 2: Recognize the Pattern Recognize that $16{�}^2$ is the square of $4�$, and $9{�}^2$ is the square of $3�$.

Step 3: Factor the Difference of Squares Factor the difference of squares using the formula $(�+�)(�-�)$: $16{�}^2-9{�}^2=(4�+3�)(4�-3�)$.

So, the factored form of $16{�}^2-9{�}^2$ is $(4�+3�)(4�-3�)$.

In both examples, recognizing the difference of squares pattern and applying the formula $(�+�)(�-�)$ allowed us to factor the expressions into binomials.

Factoring the sum and difference of cubes involves recognizing specific patterns and applying the appropriate formulas. Here's how to factor the sum and difference of cubes:

1. Factoring the Sum of Cubes:

The sum of cubes has the form ${�}^3+{�}^3$. It can be factored as follows:

${�}^3+{�}^3=(�+�)({�}^2-��+{�}^2)$

Example: Factor ${�}^3+8$.

Step 1: Identify the Sum of Cubes The given expression is ${�}^3+8$, where ${�}^3$ is the cube of $�$ and $8$ is the cube of $2$.

Step 2: Apply the Formula Apply the sum of cubes formula: ${�}^3+8=(�+2)({�}^2-2�+4)$

So, the factored form of ${�}^3+8$ is $(�+2)({�}^2-2�+4)$.

2. Factoring the Difference of Cubes:

The difference of cubes has the form ${�}^3-{�}^3$. It can be factored as follows:

${�}^3-{�}^3=(�-�)({�}^2+��+{�}^2)$

Example: Factor $8{�}^3-27$.

Step 1: Identify the Difference of Cubes The given expression is $8{�}^3-27$, where $8{�}^3$ is the cube of $2�$ and $27$ is the cube of $3$.

Step 2: Apply the Formula Apply the difference of cubes formula: $8{�}^3-27=(2�-3)(4{�}^2+6�+9)$

So, the factored form of $8{�}^3-27$ is $(2�-3)(4{�}^2+6�+9)$.

These are the formulas and steps for factoring the sum and difference of cubes. Recognizing these patterns can make factoring cubic expressions much simpler.

I'll provide examples of factoring both a sum of cubes and a difference of cubes.

1. Factoring the Sum of Cubes:

The sum of cubes formula is ${�}^3+{�}^3=(�+�)({�}^2-��+{�}^2)$.

Example: Factor ${�}^3+8$.

Step 1: Identify the Sum of Cubes In this case, ${�}^3$ is the cube of $�$ and $8$ is the cube of $2$.

Step 2: Apply the Formula Apply the sum of cubes formula: ${�}^3+8=(�+2)({�}^2-2�+4)$

So, the factored form of ${�}^3+8$ is $(�+2)({�}^2-2�+4)$.

2. Factoring the Difference of Cubes:

The difference of cubes formula is ${�}^3-{�}^3=(�-�)({�}^2+��+{�}^2)$.

Example: Factor $27{�}^3-8{�}^3$.

Step 1: Identify the Difference of Cubes In this case, $27{�}^3$ is the cube of $3�$ and $8{�}^3$ is the cube of $2�$.

Step 2: Apply the Formula Apply the difference of cubes formula: $27{�}^3-8{�}^3=(3�-2�)(9{�}^2+6��+4{�}^2)$

So, the factored form of $27{�}^3-8{�}^3$ is $(3�-2�)(9{�}^2+6��+4{�}^2)$.

These examples illustrate how to factor both a sum of cubes and a difference of cubes using the respective formulas. Recognizing these patterns can simplify factoring cubic expressions.

Factoring expressions with fractional or negative exponents follows similar principles to factoring expressions with integer exponents. However, it may require some additional steps due to the non-integer exponents. Here are some guidelines and examples:

1. Factoring Expressions with Fractional Exponents:

When factoring expressions with fractional exponents, you can use the properties of exponents to rewrite them as radicals and then factor them.

Example 1: Factor ${�}^{3\mathrm{/}2}-4{�}^{1\mathrm{/}2}$.

Step 1: Identify Common Terms Both terms have a common factor of ${�}^{1\mathrm{/}2}$, which can be factored out.

Step 2: Factor Out the Common Term Factor out ${�}^{1\mathrm{/}2}$: ${�}^{3\mathrm{/}2}-4{�}^{1\mathrm{/}2}={�}^{1\mathrm{/}2}(�-4)$

So, the factored form is ${�}^{1\mathrm{/}2}(�-4)$.

2. Factoring Expressions with Negative Exponents:

When factoring expressions with negative exponents, you can move the terms with negative exponents to the denominator and change the sign of the exponent.

Example 2: Factor $\frac{1}{{�}^{-2}{�}^3}-\frac{3}{{�}^2{�}^{-1}}$.

Step 1: Rewrite Negative Exponents Move the terms with negative exponents to the denominator and change the sign of the exponents: $\frac{{�}^3}{{�}^2}-\frac{3}{��}$.

Step 2: Identify Common Terms Both terms have a common factor of $\frac{1}{�}$, which can be factored out.

Step 3: Factor Out the Common Term Factor out $\frac{1}{�}$: $\frac{1}{�}({�}^3-3)$.

So, the factored form is $\frac{1}{�}({�}^3-3)$.

These examples demonstrate how to factor expressions with fractional and negative exponents. The key is to identify common terms, factor them out, and simplify the expression as much as possible.

Let's go through more examples of factoring expressions with fractional or negative exponents step by step.

Example 1: Factor ${�}^{2\mathrm{/}3}-8{�}^{-1\mathrm{/}3}$.

Step 1: Identify Common Terms Both terms have a common factor of ${�}^{-1\mathrm{/}3}$, which can be factored out.

Step 2: Factor Out the Common Term Factor out ${�}^{-1\mathrm{/}3}$: ${�}^{2\mathrm{/}3}-8{�}^{-1\mathrm{/}3}={�}^{-1\mathrm{/}3}(�-8)$

So, the factored form is ${�}^{-1\mathrm{/}3}(�-8)$.

Example 2: Factor $\frac{1}{{�}^{-2\mathrm{/}5}{�}^{3\mathrm{/}4}}-\frac{3}{{�}^{2\mathrm{/}5}{�}^{-1\mathrm{/}4}}$.

Step 1: Rewrite Negative Exponents Move the terms with negative exponents to the denominator and change the sign of the exponents: $\frac{{�}^{3\mathrm{/}4}}{{�}^{2\mathrm{/}5}}-\frac{3}{�{�}^{1\mathrm{/}4}}$.

Step 2: Identify Common Terms Both terms have a common factor of $\frac{1}{{�}^{2\mathrm{/}5}}$, which can be factored out.

Step 3: Factor Out the Common Term Factor out $\frac{1}{{�}^{2\mathrm{/}5}}$: $\frac{1}{{�}^{2\mathrm{/}5}}({�}^{3\mathrm{/}4}-3{�}^{1\mathrm{/}4})$.

So, the factored form is $\frac{1}{{�}^{2\mathrm{/}5}}({�}^{3\mathrm{/}4}-3{�}^{1\mathrm{/}4})$.

Example 3: Factor ${�}^{-1}{�}^{-2}+4{�}^{1\mathrm{/}2}{�}^{-3}$.

Step 1: Identify Common Terms Both terms have a common factor of ${�}^{-2}$, which can be factored out.

Step 2: Factor Out the Common Term Factor out ${�}^{-2}$: ${�}^{-1}{�}^{-2}+4{�}^{1\mathrm{/}2}{�}^{-3}={�}^{-2}({�}^{-1}+4{�}^{1\mathrm{/}2})$

So, the factored form is ${�}^{-2}({�}^{-1}+4{�}^{1\mathrm{/}2})$.

These examples illustrate how to factor expressions with fractional and negative exponents by identifying common terms and factoring them out. Remember to use the properties of exponents to simplify the expressions as needed.

Here are three more examples of factoring expressions with fractional or negative exponents along with their answers:

Example 4: Factor ${�}^{1\mathrm{/}4}-2{�}^{-3\mathrm{/}4}$.

Step 1: Identify Common Terms Both terms have a common factor of ${�}^{-3\mathrm{/}4}$, which can be factored out.

Step 2: Factor Out the Common Term Factor out ${�}^{-3\mathrm{/}4}$: ${�}^{1\mathrm{/}4}-2{�}^{-3\mathrm{/}4}={�}^{-3\mathrm{/}4}(�+2)$

So, the factored form is ${�}^{-3\mathrm{/}4}(�+2)$.

Example 5: Factor $\frac{1}{{�}^{1\mathrm{/}3}{�}^{-2}}-\frac{2}{{�}^{2\mathrm{/}3}{�}^{-1}}$.

Step 1: Rewrite Negative Exponents Move the terms with negative exponents to the denominator and change the sign of the exponents: $\frac{{�}^2}{{�}^{1\mathrm{/}3}}-\frac{2�}{�}$.

Step 2: Identify Common Terms Both terms have a common factor of $\frac{1}{�}$, which can be factored out.

Step 3: Factor Out the Common Term Factor out $\frac{1}{�}$: $\frac{1}{�}({�}^2{�}^{1\mathrm{/}3}-2�)$.

So, the factored form is $\frac{1}{�}({�}^2{�}^{1\mathrm{/}3}-2�)$.

Example 6: Factor ${�}^{-1\mathrm{/}2}+9{�}^{3\mathrm{/}2}$.

Step 1: Identify Common Terms Both terms have a common factor of ${�}^{-1\mathrm{/}2}$, which can be factored out.

Step 2: Factor Out the Common Term Factor out ${�}^{-1\mathrm{/}2}$: ${�}^{-1\mathrm{/}2}+9{�}^{3\mathrm{/}2}={�}^{-1\mathrm{/}2}(1+9{�}^2)$

So, the factored form is ${�}^{-1\mathrm{/}2}(1+9{�}^2)$.

These additional examples should help you practice factoring expressions with fractional or negative exponents. The key is to recognize common terms and factor them out to simplify the expression.