Q1: Find the quotient and remainder using the specified method.
a) 2๐ฅ^3 โ 7๐ฅ โ 9 divided by ๐ฅ2 + 2๐ฅ + 1 using long division.
b) 6๐ฅ^5 โ ๐ฅ^2 โ 4๐ฅ +3 divided by ๐ฅ + 2 using synthetic division.
such divisions are almost impossible to format legibly on this web site.
But a few seconds online will produce handy calculators that show all the steps.
can u please tell me some other website it would be really helpful to me!
I'll give you one.
Surely you can make the effort to find the other.
www.emathhelp.net/en/calculators/algebra-1/polynomial-long-division-calculator/
Thank you a lot!
To find the quotient and remainder using the specified methods (long division and synthetic division), follow these steps:
a) Long Division:
Step 1: Arrange the dividend and divisor in descending order of powers of x.
Dividend: 2๐ฅ^3 โ 7๐ฅ โ 9
Divisor: ๐ฅ^2 + 2๐ฅ + 1
Step 2: Divide the first term of the dividend by the first term of the divisor to get the first term of the quotient.
Quotient = 2๐ฅ
Step 3: Multiply the divisor by the obtained term of the quotient.
Product = 2๐ฅ * (๐ฅ^2 + 2๐ฅ + 1) = 2๐ฅ^3 + 4๐ฅ^2 + 2๐ฅ
Step 4: Subtract the product from the dividend.
Remainder = (2๐ฅ^3 โ 7๐ฅ โ 9) - (2๐ฅ^3 + 4๐ฅ^2 + 2๐ฅ) = -4๐ฅ^2 - 9๐ฅ - 9
Therefore, the quotient is 2๐ฅ and the remainder is -4๐ฅ^2 - 9๐ฅ - 9.
b) Synthetic Division:
Step 1: Set up synthetic division using the divisor and the coefficients of the dividend.
Dividend: 6๐ฅ^5 โ ๐ฅ^2 โ 4๐ฅ + 3
Divisor: ๐ฅ + 2
Step 2: Write the coefficients of the dividend, including zeros for missing terms.
6 0 -1 -4 0 3
Step 3: Use synthetic division to divide the first coefficient by the divisor:
-2 | 6 0 -1 -4 0 3
-12 24 -46 100 -200
---------------------------------------
6 -12 23 96 100 -197
The top row of the result gives the coefficients of the quotient: 6 - 12๐ฅ + 23๐ฅ^2 + 96๐ฅ^3 + 100๐ฅ^4 - 197๐ฅ^5.
The last value in the bottom row (-197) represents the remainder.
Therefore, the quotient is 6 - 12๐ฅ + 23๐ฅ^2 + 96๐ฅ^3 + 100๐ฅ^4 - 197๐ฅ^5 and the remainder is -197.