(6x^3-8x^2-14x+13)/(3x+2)
To divide the polynomial (6x^3 - 8x^2 - 14x + 13) by (3x + 2), you can use long division.
1. Write the dividend (6x^3 - 8x^2 - 14x + 13) inside the division symbol.
2. Write the divisor (3x + 2) outside the division symbol.
3. Divide the first term of the dividend (6x^3) by the first term of the divisor (3x), which gives 2x^2. Write this as the first term of the quotient above the division symbol.
4. Multiply the entire divisor (3x + 2) by the first term of the quotient (2x^2), which gives 6x^3 + 4x^2. Write this result below the dividend aligned with the appropriate terms.
5. Subtract the result from step 4 (6x^3 + 4x^2) from the original dividend (6x^3 - 8x^2 - 14x + 13). This will cancel the highest degree term (or cancel multiple terms if necessary) and bring down the next term (-14x).
(6x^3 - 8x^2 - 14x + 13) - (6x^3 + 4x^2) = -12x^2 - 14x + 13
6. Repeat the process by dividing the first term of the new dividend (-12x^2) by the first term of the divisor (3x), which gives -4x. Write this as the second term of the quotient above the division symbol.
7. Multiply the entire divisor (3x + 2) by the second term of the quotient (-4x), which gives -12x^2 - 8x. Write this result below the difference obtained in step 5 aligned with the appropriate terms.
8. Subtract the result from step 7 (-12x^2 - 8x) from the difference obtained in step 5 (-12x^2 - 14x + 13). This will cancel the highest degree term (or cancel multiple terms if necessary) and bring down the next term (13).
(-12x^2 - 14x + 13) - (-12x^2 - 8x) = -6x - 7
9. Repeat the process by dividing the first term of the new dividend (-6x) by the first term of the divisor (3x), which gives -2. Write this as the third term of the quotient above the division symbol.
10. Multiply the entire divisor (3x + 2) by the third term of the quotient (-2), which gives -6x - 4. Write this result below the difference obtained in step 8 aligned with the appropriate terms.
11. Subtract the result from step 10 (-6x - 4) from the difference obtained in step 8 (-6x - 7). This will cancel the highest degree term (or cancel multiple terms if necessary) and leave no remainder.
(-6x - 7) - (-6x - 4) = -3
12. Since we have no remainder, the division is complete. The quotient is 2x^2 - 4x - 2, and there is no remainder.
Therefore, the result of (6x^3 - 8x^2 - 14x + 13) divided by (3x + 2) is:
Quotient: 2x^2 - 4x - 2
Remainder: 0