solve (3v+4)/(v-2)
To solve the expression (3v+4)/(v-2), you can simplify it by performing polynomial division or by using the method of partial fractions.
Method 1: Polynomial Division
1. Start by dividing the numerator (3v+4) by the denominator (v-2) using long division:
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v-2 | 3v + 4
2. Divide the first term of the numerator (3v) by the first term of the denominator (v) to get 3. Write this result on top.
3
3. Multiply this result (3) by the entire denominator (v-2) and write it below the numerator.
3v - 6
4. Subtract this product (3v-6) from the numerator (3v+4):
(3v + 4) - (3v - 6)
Simplify:
3v + 4 - 3v + 6 = 10
5. Write this result (10) as the new numerator:
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v-2 | 3v + 4
10
6. Since the degree of the new numerator (10) is less than the degree of the denominator (v-2), the division is complete.
Therefore, the simplified form of (3v+4)/(v-2) is 3 + 10/(v-2).
Method 2: Partial Fractions
In this method, we decompose the expression (3v+4)/(v-2) into simpler fractions.
1. Start by factoring the denominator (v-2) if possible. In this case, it is already in factored form.
2. Write the expression as a sum of two fractions with unknown numerators and the factored denominator:
(3v+4)/(v-2) = A/(v-2) + B
3. Multiply both sides of the equation by the denominator (v-2) to eliminate the denominators:
(3v+4) = A + B(v-2)
4. Substitute a value for v that makes one of the terms disappear. Let's choose v=2, which makes the term (v-2) equal to 0.
Thus, we get:
(3(2)+4) = A + B(2-2)
10 = A
5. Now substitute this value of A back into the equation and solve for B:
(3v+4)/(v-2) = 10/(v-2) + B
To solve for B, we can substitute another value for v. Let's choose v=3:
(3(3)+4)/(3-2) = 10/(3-2) + B
13 = 10 + B
B = 13 - 10
B = 3
6. Now we know that A=10 and B=3.
Therefore, the simplified form of (3v+4)/(v-2) is 10/(v-2) + 3.
Note: Both methods yield the same simplified expression.