%C3%B7(2m-1).jpeg "(2m^3+m^2+m+9)÷(2m-1)")
Polynomial Long Division: (2m^3 + m^2 + m + 9) ÷ (2m - 1)
This article will guide you through the process of dividing the polynomial (2m^3 + m^2 + m + 9) by (2m - 1) using polynomial long division.
Steps:
-
Set up the division problem:
________ 2m-1 | 2m^3 + m^2 + m + 9 -
Divide the leading terms:
- Divide the leading term of the dividend (2m^3) by the leading term of the divisor (2m): (2m^3) / (2m) = m^2
- Write m^2 above the m^2 term in the dividend.
m^2_______ 2m-1 | 2m^3 + m^2 + m + 9 -
Multiply the divisor by the quotient term:
- Multiply (2m - 1) by m^2: (2m - 1)(m^2) = 2m^3 - m^2
- Write the result below the dividend, aligning like terms:
m^2_______ 2m-1 | 2m^3 + m^2 + m + 9 2m^3 - m^2 ------- -
Subtract:
- Subtract the terms: (2m^3 + m^2) - (2m^3 - m^2) = 2m^2
- Bring down the next term from the dividend:
m^2_______ 2m-1 | 2m^3 + m^2 + m + 9 2m^3 - m^2 ------- 2m^2 + m -
Repeat steps 2-4:
- Divide the leading term of the new dividend (2m^2) by the leading term of the divisor (2m): (2m^2) / (2m) = m
- Write m above the m term in the dividend.
m^2 + m____ 2m-1 | 2m^3 + m^2 + m + 9 2m^3 - m^2 ------- 2m^2 + m- Multiply (2m - 1) by m: (2m - 1)(m) = 2m^2 - m
- Write the result below:
m^2 + m____ 2m-1 | 2m^3 + m^2 + m + 9 2m^3 - m^2 ------- 2m^2 + m 2m^2 - m -------- Subtract: (2m^2 + m) - (2m^2 - m) = 2m
- Bring down the next term:
m^2 + m____ 2m-1 | 2m^3 + m^2 + m + 9 2m^3 - m^2 ------- 2m^2 + m 2m^2 - m ------- 2m + 9 -
Repeat steps 2-4 again:
- Divide the leading term of the new dividend (2m) by the leading term of the divisor (2m): (2m) / (2m) = 1
- Write 1 above the constant term in the dividend.
m^2 + m + 1_ 2m-1 | 2m^3 + m^2 + m + 9 2m^3 - m^2 ------- 2m^2 + m 2m^2 - m ------- 2m + 9 2m - 1 ------ Multiply (2m - 1) by 1: (2m - 1)(1) = 2m - 1
- Write the result below:
m^2 + m + 1_ 2m-1 | 2m^3 + m^2 + m + 9 2m^3 - m^2 ------- 2m^2 + m 2m^2 - m ------- 2m + 9 2m - 1 ------ Subtract: (2m + 9) - (2m - 1) = 10
m^2 + m + 1_ 2m-1 | 2m^3 + m^2 + m + 9 2m^3 - m^2 ------- 2m^2 + m 2m^2 - m ------- 2m + 9 2m - 1 ----- 10 -
The Result:
- The quotient is m^2 + m + 1.
- The remainder is 10.
Therefore, (2m^3 + m^2 + m + 9) ÷ (2m - 1) = m^2 + m + 1 + 10/(2m - 1).