How to solve (6x^4+30x^3+3x+14) / (x+) in long division. I understand synthetic but not long divison
Long division for polynomials involves dividing each term of the polynomial by the divisor, similar to long division with numbers. Here are the steps to solve (6x^4+30x^3+3x+14) / (x+):
1. Write the dividend and divisor in long division format, with the dividend on top and the divisor on the outside to the left.
_______
x+ |6x^4+30x^3+3x+14
2. Divide the first term of the dividend by the first term of the divisor, and write the quotient above the dividend.
6x^3
_______
x+ |6x^4+30x^3+3x+14
3. Multiply the divisor (x+) by the quotient (6x^3) and write the result underneath the first two terms of the dividend.
6x^3
_______
x+ |6x^4+30x^3+3x+14
6x^4+6x^3
4. Subtract the product from step 3 from the first two terms of the dividend, and bring down the next term of the dividend.
6x^3
_______
x+ |6x^4+30x^3+3x+14
6x^4+6x^3
_________
24x^3 + 3x
5. Repeat steps 2-4 with the new dividend (24x^3+3x).
6x^3 + 24x^2
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x+ |6x^4+30x^3+3x+14
6x^4+6x^3
_________
24x^3 + 3x
24x^3 + 24x^2
_____________
-21x^2 + 3x
6. Repeat steps 2-4 with the new dividend (-21x^2+3x).
6x^3 + 24x^2 - 21x
_______
x+ |6x^4+30x^3+3x+14
6x^4+6x^3
_________
24x^3 + 3x
24x^3 + 24x^2
_____________
-21x^2 + 3x
-21x^2 - 21x
_____________
24
7. The quotient is 6x^3 + 24x^2 - 21x with a remainder of 24, written as:
6x^3 + 24x^2 - 21x + (24 / (x+))
To solve the division problem (6x^4+30x^3+3x+14) divided by (x+), we will use long division. Here are the step-by-step instructions:
Step 1: Arrange the terms in descending order of powers of x. If any term is missing, write it with a coefficient of zero.
6x^4 + 30x^3 + 0x^2 + 3x + 14
Step 2: Divide the first term of the dividend (6x^4) by the first term of the divisor (x). Write the result above the line.
_____________________
x + | 6x^4 + 30x^3 + 0x^2 + 3x + 14
-
6x^3
Step 3: Multiply the entire divisor (x+) by the result from Step 2 (6x^3). Write the result below the subtraction line.
_____________________
x + | 6x^4 + 30x^3 + 0x^2 + 3x + 14
- (6x^3 + )
6x^4 +
Step 4: Subtract the two polynomials. Bring down the next term from the dividend.
_____________________
x + | 6x^4 + 30x^3 + 0x^2 + 3x + 14
- (6x^3 + )
6x^4 +
- (6x^4 + 30x^3)
0x^3 +
Step 5: Repeat steps 2-4 until all terms have been processed.
_____________________
x + | 6x^4 + 30x^3 + 0x^2 + 3x + 14
- (6x^3 + )
6x^4 +
- (6x^4 + 30x^3)
0x^3 +
- (0x^3 + 0x^2)
0x^2 +
- (0x^2 + 3x)
- 3x +
- ( - 3x + 14)
11
Step 6: The final result is the quotient polynomial plus the remainder, if any. In this case, the quotient is 6x^3 + 0x^2 - 3x - 3, and there is a remainder of 11.
Therefore, (6x^4+30x^3+3x+14) divided by (x+) equals 6x^3 + 0x^2 - 3x - 3 with a remainder of 11.