(z^3+3z^2+2z+6)/(z^2+5z+6)
I don't know what to do??????
To simplify the given expression, you need to perform polynomial division. Here are the steps:
Step 1: Make sure the expression is in descending order of powers. In this case, it already is.
Step 2: Identify the quotient and the remainder you want to find. You are dividing (z^3 + 3z^2 + 2z + 6) by (z^2 + 5z + 6).
Step 3: Start by dividing the highest power terms. Divide z^3 by z^2, which gives you z. Write this as the first term of the quotient.
Step 4: Multiply the entire divisor (z^2 + 5z + 6) by the first term of the quotient (z). Then subtract the result from the original expression to cancel out the z^3 term.
(z^2 + 5z + 6) multiplied by z is z^3 + 5z^2 + 6z.
(z^3 + 3z^2 + 2z + 6) - (z^3 + 5z^2 + 6z) = -2z^2 -4z + 6.
Step 5: Repeat step 3 with the next highest power term in the remainder. Divide -2z^2 by z^2, which gives -2. Write this as the next term of the quotient.
Step 6: Multiply the entire divisor (z^2 + 5z + 6) by the new term of the quotient (-2). Then subtract the result from the previous remainder to cancel out the -2z^2 term.
(z^2 + 5z + 6) multiplied by -2 is -2z^2 - 10z - 12.
(-2z^2 -4z + 6) - (-2z^2 - 10z - 12) = 6z + 18.
Step 7: Repeat step 3 with the next highest power term in the new remainder. Divide 6z by z^2, which gives 0. Write this as the next term of the quotient.
Step 8: Multiply the entire divisor (z^2 + 5z + 6) by the new term of the quotient (0). Then subtract the result from the previous remainder to cancel out the 6z term.
(z^2 + 5z + 6) multiplied by 0 is 0.
(6z + 18) - 0 = 6z + 18.
Step 9: Since there are no more terms left in the new remainder, the division stops here.
The final result is:
Quotient: z - 2
Remainder: 6z + 18
So, the simplified expression is:
(z^3 + 3z^2 + 2z + 6) / (z^2 + 5z + 6) = z - 2 + (6z + 18) / (z^2 + 5z + 6)