SOLUTION: Divide and simplify. (z^2-1/4z+4)/(z-1/2)

Question 268637: Divide and simplify. (z^2-1/4z+4)/(z-1/2)
Answer by persian52(161) (Show Source): You can put this solution on YOUR website!
(z^(2)-(1)/(4)*z+4)/(z-(1)/(2))
►Multiply -(1)/(4) by z to get -(z)/(4).
(z^(2)-(z)/(4)+4)/(z-(1)/(2))
►To add fractions, the denominators must be equal. The denominators can be made equal by finding the least common denominator (LCD). In this case, the LCD is 4. Next, multiply each fraction by a factor of 1 that will create the LCD in each of the fractions.
z^(2)*(4)/(4)-(z)/(4)+4*(4)/(4)/(z-(1)/(2))
►Complete the multiplication to produce a denominator of 4 in each expression.
(4z^(2))/(4)-(z)/(4)+(16)/(4)/(z-(1)/(2))
►Combine the numerators of all expressions that have common denominators.
(4z^(2)+16)/(4)-(z)/(4)/(z-(1)/(2))
►To add fractions, the denominators must be equal. The denominators can be made equal by finding the least common denominator (LCD). In this case, the LCD is 4. Next, multiply each fraction by a factor of 1 that will create the LCD in each of the fractions.
z^(2)*(4)/(4)-(z)/(4)+4*(4)/(4)/(z-(1)/(2))
►Complete the multiplication to produce a denominator of 4 in each expression.
(4z^(2))/(4)-(z)/(4)+(16)/(4)/(z-(1)/(2))
►Combine the numerators of all expressions that have common denominators.
(4z^(2)+16)/(4)-(z)/(4)/(z-(1)/(2))
►Factor out the GCF of 4 from each term in the polynomial.
(4(z^(2))+4(4))/(4)-(z)/(4)/(z-(1)/(2))
►Factor out the GCF of 4 from 4z^(2)+16.
(4(z^(2)+4))/(4)-(z)/(4)/(z-(1)/(2))
►Reduce the expression (4(z^(2)+4))/(4) by removing a factor of 4 from the numerator and denominator.
(z^(2)+4)-(z)/(4)/(z-(1)/(2))
►Remove the parentheses around the expression z^(2)+4.
z^(2)+4-(z)/(4)/(z-(1)/(2))
►To add fractions, the denominators must be equal. The denominators can be made equal by finding the least common denominator (LCD). In this case, the LCD is 2. Next, multiply each fraction by a factor of 1 that will create the LCD in each of the fractions.
z^(2)-(z)/(4)+(4)/(z*(2)/(2)-(1)/(2))
►Complete the multiplication to produce a denominator of 2 in each expression.
z^(2)-(z)/(4)+(4)/((2z)/(2)-(1)/(2))
►Combine the numerators of all expressions that have common denominators.
z^(2)-(z)/(4)+(4)/((2z-1)/(2))
►Any number raised to the 1st power is the number.
z^(2)-(z)/(4)+(4)/(((1)/(2))(2z-1))
►Multiply 4 by 2 to get 8.
z^(2)-(z)/(4)+(8)/(2z-1)
►To add fractions, the denominators must be equal. The denominators can be made equal by finding the least common denominator (LCD). In this case, the LCD is 4. Next, multiply each fraction by a factor of 1 that will create the LCD in each of the fractions.
z^(2)*(4)/(4)-(z)/(4)+(8)/(2z-1)
►Complete the multiplication to produce a denominator of 4 in each expression.
(4z^(2))/(4)-(z)/(4)+(8)/(2z-1)
►Reduce the expression (4z^(2))/(4) by removing a factor of 4 from the numerator and denominator.
z^(2)-(z)/(4)+(8)/(2z-1)
►Multiply each term by a factor of 1 that will equate all the denominators. In this case, all terms need a denominator of 4(2z-1). The -(z)/(4) expression needs to be multiplied by ((2z-1))/((2z-1)) to make the denominator 4(2z-1). The (8)/((2z-1)) expression needs to be multiplied by ((4))/((4)) to make the denominator 4(2z-1).
z^(2)*(4(2z-1))/(4(2z-1))-(z)/(4)*(2z-1)/(2z-1)+(8)/(2z-1)*(4)/(4)
►Multiply the expression by a factor of 1 to create the least common denominator (LCD) of 4(2z-1).
(z^(2)*4(2z-1))/(4(2z-1))-(z)/(4)*(2z-1)/(2z-1)+(8)/(2z-1)*(4)/(4)
►Multiply z^(2) by 4 to get 4z^(2).
(4z^(2)(2z-1))/(4(2z-1))-(z)/(4)*(2z-1)/(2z-1)+(8)/(2z-1)*(4)/(4)
►Multiply the expression by a factor of 1 to create the least common denominator (LCD) of 4(2z-1).
(4z^(2)(2z-1))/(4(2z-1))-(z(2z-1))/(4(2z-1))+(8)/(2z-1)*(4)/(4)
►Multiply the expression by a factor of 1 to create the least common denominator (LCD) of 4(2z-1).
(4z^(2)(2z-1))/(4(2z-1))-(z(2z-1))/(4(2z-1))+(8(4))/(4(2z-1))
►The numerators of expressions that have equal denominators can be combined. In this case, ((4z^(2)(2z-1)))/(4(2z-1)) and -((z(2z-1)))/(4(2z-1)) have the same denominator of 4(2z-1), so the numerators can be combined.
((4z^(2)(2z-1))-(z(2z-1))+(8(4)))/(4(2z-1))
►Simplify the numerator of the expression.
=► (8z^(3)-6z^(2)+z+32)/(4(2z-1))
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