Join the IDNLearn.com community and start exploring a world of knowledge today. Receive prompt and accurate responses to your questions from our community of knowledgeable professionals ready to assist you at any time.
Sagot :
Sure, let's divide the given rational expressions step by step. The expression that we need to evaluate is:
[tex]\[ \frac{6 s^2 + 7 s t - 5 t^2}{6 s^2 - 5 s t + t^2} \div \frac{3 s^2 + 26 s t + 35 t^2}{6 s^2 + 19 s t - 7 t^2} \][/tex]
Dividing by a fraction is the same as multiplying by its reciprocal. So we can rewrite the problem as:
[tex]\[ \frac{6 s^2 + 7 s t - 5 t^2}{6 s^2 - 5 s t + t^2} \times \frac{6 s^2 + 19 s t - 7 t^2}{3 s^2 + 26 s t + 35 t^2} \][/tex]
Now, we multiply the two fractions together by multiplying the numerators and the denominators:
### Step 1: Multiply the numerators
[tex]\[ (6 s^2 + 7 s t - 5 t^2) \times (6 s^2 + 19 s t - 7 t^2) \][/tex]
### Step 2: Multiply the denominators
[tex]\[ (6 s^2 - 5 s t + t^2) \times (3 s^2 + 26 s t + 35 t^2) \][/tex]
### Expanded Form
Let's expand these expressions:
Numerator Expansion:
[tex]\[ (6 s^2 + 7 s t - 5 t^2)(6 s^2 + 19 s t - 7 t^2) = 6 s^2 (6 s^2 + 19 s t - 7 t^2) + 7 s t (6 s^2 + 19 s t - 7 t^2) - 5 t^2 (6 s^2 + 19 s t - 7 t^2) \][/tex]
Simplify:
[tex]\[ = 36 s^4 + 114 s^3 t - 42 s^2 t^2 + 42 s^3 t + 133 s^2 t^2 - 49 s t^3 - 30 s^2 t^2 - 95 s t^3 + 35 t^4 \][/tex]
Combine like terms:
[tex]\[ = 36 s^4 + 156 s^3 t + 61 s^2 t^2 - 144 s t^3 + 35 t^4 \][/tex]
Denominator Expansion:
[tex]\[ (6 s^2 - 5 s t + t^2)(3 s^2 + 26 s t + 35 t^2) = 6 s^2 (3 s^2 + 26 s t + 35 t^2) - 5 s t (3 s^2 + 26 s t + 35 t^2) + t^2 (3 s^2 + 26 s t + 35 t^2) \][/tex]
Simplify:
[tex]\[ = 18 s^4 + 156 s^3 t + 210 s^2 t^2 - 15 s^3 t - 130 s^2 t^2 - 175 s t^3 + 3 s^2 t^2 + 26 s t^3 + 35 t^4 \][/tex]
Combine like terms:
[tex]\[ = 18 s^4 + 141 s^3 t + 83 s^2 t^2 - 149 s t^3 + 35 t^4 \][/tex]
### Simplify the Expression
Now we have:
[tex]\[ \frac{36 s^4 + 156 s^3 t + 61 s^2 t^2 - 144 s t^3 + 35 t^4}{18 s^4 + 141 s^3 t + 83 s^2 t^2 - 149 s t^3 + 35 t^4} \][/tex]
Finally, this fraction can sometimes be simplified further, which is often done by factoring common elements out from the numerator and the denominator. However, without specific factors or more detailed simplification steps, we keep it in its expanded form.
Thus, our resulting rational expression is:
[tex]\[ \frac{36 s^4 + 156 s^3 t + 61 s^2 t^2 - 144 s t^3 + 35 t^4}{18 s^4 + 141 s^3 t + 83 s^2 t^2 - 149 s t^3 + 35 t^4} \][/tex]
This is the simplified form of the given division of rational expressions.
[tex]\[ \frac{6 s^2 + 7 s t - 5 t^2}{6 s^2 - 5 s t + t^2} \div \frac{3 s^2 + 26 s t + 35 t^2}{6 s^2 + 19 s t - 7 t^2} \][/tex]
Dividing by a fraction is the same as multiplying by its reciprocal. So we can rewrite the problem as:
[tex]\[ \frac{6 s^2 + 7 s t - 5 t^2}{6 s^2 - 5 s t + t^2} \times \frac{6 s^2 + 19 s t - 7 t^2}{3 s^2 + 26 s t + 35 t^2} \][/tex]
Now, we multiply the two fractions together by multiplying the numerators and the denominators:
### Step 1: Multiply the numerators
[tex]\[ (6 s^2 + 7 s t - 5 t^2) \times (6 s^2 + 19 s t - 7 t^2) \][/tex]
### Step 2: Multiply the denominators
[tex]\[ (6 s^2 - 5 s t + t^2) \times (3 s^2 + 26 s t + 35 t^2) \][/tex]
### Expanded Form
Let's expand these expressions:
Numerator Expansion:
[tex]\[ (6 s^2 + 7 s t - 5 t^2)(6 s^2 + 19 s t - 7 t^2) = 6 s^2 (6 s^2 + 19 s t - 7 t^2) + 7 s t (6 s^2 + 19 s t - 7 t^2) - 5 t^2 (6 s^2 + 19 s t - 7 t^2) \][/tex]
Simplify:
[tex]\[ = 36 s^4 + 114 s^3 t - 42 s^2 t^2 + 42 s^3 t + 133 s^2 t^2 - 49 s t^3 - 30 s^2 t^2 - 95 s t^3 + 35 t^4 \][/tex]
Combine like terms:
[tex]\[ = 36 s^4 + 156 s^3 t + 61 s^2 t^2 - 144 s t^3 + 35 t^4 \][/tex]
Denominator Expansion:
[tex]\[ (6 s^2 - 5 s t + t^2)(3 s^2 + 26 s t + 35 t^2) = 6 s^2 (3 s^2 + 26 s t + 35 t^2) - 5 s t (3 s^2 + 26 s t + 35 t^2) + t^2 (3 s^2 + 26 s t + 35 t^2) \][/tex]
Simplify:
[tex]\[ = 18 s^4 + 156 s^3 t + 210 s^2 t^2 - 15 s^3 t - 130 s^2 t^2 - 175 s t^3 + 3 s^2 t^2 + 26 s t^3 + 35 t^4 \][/tex]
Combine like terms:
[tex]\[ = 18 s^4 + 141 s^3 t + 83 s^2 t^2 - 149 s t^3 + 35 t^4 \][/tex]
### Simplify the Expression
Now we have:
[tex]\[ \frac{36 s^4 + 156 s^3 t + 61 s^2 t^2 - 144 s t^3 + 35 t^4}{18 s^4 + 141 s^3 t + 83 s^2 t^2 - 149 s t^3 + 35 t^4} \][/tex]
Finally, this fraction can sometimes be simplified further, which is often done by factoring common elements out from the numerator and the denominator. However, without specific factors or more detailed simplification steps, we keep it in its expanded form.
Thus, our resulting rational expression is:
[tex]\[ \frac{36 s^4 + 156 s^3 t + 61 s^2 t^2 - 144 s t^3 + 35 t^4}{18 s^4 + 141 s^3 t + 83 s^2 t^2 - 149 s t^3 + 35 t^4} \][/tex]
This is the simplified form of the given division of rational expressions.
We appreciate your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. Your questions are important to us at IDNLearn.com. Thanks for stopping by, and come back for more reliable solutions.
