Question

remainders quick check divide the polynomial 12x^2 + 2x -9 by 4x-2

Answer 1

To divide the polynomial \(12x^2 + 2x - 9\) by \(4x - 2\), we can use polynomial long division.

1. **Set up the long division**: Write \(12x^2 + 2x - 9\) under the long division symbol and \(4x - 2\) outside.

2. **Determine the first term of the quotient**: Divide the leading term \(12x^2\) by the leading term \(4x\):
\[
\frac{12x^2}{4x} = 3x
\]
So, the first term of the quotient is \(3x\).

3. **Multiply**: Multiply \(3x\) by \(4x - 2\):
\[
3x(4x - 2) = 12x^2 - 6x
\]

4. **Subtract**: Subtract \(12x^2 - 6x\) from \(12x^2 + 2x - 9\):
\[
(12x^2 + 2x - 9) - (12x^2 - 6x) = 2x - 9 + 6x = 8x - 9
\]

5. **Repeat the process**: Now, divide the leading term \(8x\) by the leading term \(4x\):
\[
\frac{8x}{4x} = 2
\]
So, the next term of the quotient is \(2\).

6. **Multiply**: Multiply \(2\) by \(4x - 2\):
\[
2(4x - 2) = 8x - 4
\]

7. **Subtract**: Subtract \(8x - 4\) from \(8x - 9\):
\[
(8x - 9) - (8x - 4) = -9 + 4 = -5
\]

Now, we have a remainder of \(-5\).

Putting it all together, we have:
\[
12x^2 + 2x - 9 = (4x - 2)(3x + 2) - 5
\]

Thus, the result of dividing \(12x^2 + 2x - 9\) by \(4x - 2\) is:
\[
\text{Quotient: } 3x + 2, \text{ Remainder: } -5
\]