what is the quotient when 3x^3-11x^2+10x-12 is divided by x-3 ?
You can use long division method or you can use Horner method.
Horner is easier and faster.
3 | 3 -11. 10 -12
| 9 -6 12
---------------
3 -2 4 0
So, the quotient is 3x^2 - 2x + 4
To find the quotient when 3x^3 - 11x^2 + 10x - 12 is divided by x - 3, you can use polynomial long division. Here are the steps:
Step 1: Write the dividend (3x^3 - 11x^2 + 10x - 12) and the divisor (x - 3).
Step 2: Divide the first term of the dividend (3x^3) by the first term of the divisor (x). This gives you 3x^2.
Step 3: Multiply the divisor (x - 3) by the quotient obtained in Step 2 (3x^2). This gives you 3x^3 - 9x^2.
Step 4: Subtract the product obtained in Step 3 (3x^3 - 9x^2) from the dividend (3x^3 - 11x^2 + 10x - 12). This gives you -2x^2 + 10x - 12.
Step 5: Bring down the next term from the dividend, which is 10x.
Step 6: Divide the first term of the new expression (-2x^2 + 10x) by the first term of the divisor (x). This gives you -2x.
Step 7: Multiply the divisor (x - 3) by the quotient obtained in Step 6 (-2x). This gives you -2x^2 + 6x.
Step 8: Subtract the product obtained in Step 7 (-2x^2 + 6x) from the new expression (-2x^2 + 10x - 12). This gives you 4x - 12.
Step 9: Bring down the next term from the dividend, which is -12.
Step 10: Divide the first term of the new expression (4x - 12) by the first term of the divisor (x). This gives you 4.
Step 11: Multiply the divisor (x - 3) by the quotient obtained in Step 10 (4). This gives you 4x - 12.
Step 12: Subtract the product obtained in Step 11 (4x - 12) from the new expression (4x - 12). This gives you a remainder of 0.
Therefore, the quotient when 3x^3 - 11x^2 + 10x - 12 is divided by x - 3 is 3x^2 - 2x + 4.
To find the quotient when a polynomial is divided by another polynomial, we can use the polynomial long division method.
Step 1: Arrange the polynomial in descending order of powers. The dividend is 3x^3 - 11x^2 + 10x - 12, and the divisor is x - 3.
3x^3 - 11x^2 + 10x - 12
______________
Step 2: Divide the first term of the dividend by the first term of the divisor. In this case, 3x^3 / x = 3x^2. Write this result above the division line.
3x^2
____________
x - 3 | 3x^3 - 11x^2 + 10x - 12
Step 3: Multiply the divisor (x - 3) by the quotient (3x^2) and write the result below the dividend, making sure to align like terms.
3x^2
____________
x - 3 | 3x^3 - 11x^2 + 10x - 12
- (3x^3 - 9x^2)
_________________
-2x^2 + 10x
Step 4: Bring down the next term (10x) and continue dividing. Repeat the process until there are no more terms left.
3x^2 + (-2x + 10)
___________________
x - 3 | 3x^3 - 11x^2 + 10x - 12
- (3x^3 - 9x^2)
_________________
-2x^2 + 10x - 12
- (-2x^2 + 6x)
___________________
4x - 12
- (4x - 12)
______________
0
Step 5: The quotient is the sum of the terms above the division line: 3x^2 + (-2x + 10) + (4x - 12). Simplifying, we get the quotient as:
3x^2 - 2x + 4
Therefore, the quotient when 3x^3 - 11x^2 + 10x - 12 is divided by x - 3 is 3x^2 - 2x + 4.