Divide the polynomial 12x^2+2x−9 by 4x−2 . Then express the polynomial as the product of the divisor and the quotient plus the remainder. (1 point)
1. 12x^2+2x-9=(4x-2)(3x+2)-13
2. 12x^2+2x-9=(4x-2)(3x+2)+5
3. 12x^2+2x-9=(4x-2)(3x+2)-5
4. 12x^2+2x-9=(4x-2)(3x-2)-11
The correct answer is:
2. 12x^2+2x-9=(4x-2)(3x+2)+5
The correct answer is:
1. 12x^2+2x-9=(4x-2)(3x+2)-13
To divide the polynomial 12x^2 + 2x - 9 by 4x - 2, you need to perform polynomial long division. Here's a step-by-step explanation on how to do it:
1. Start by writing the divisor (4x - 2) on the left side of a long division symbol (like the division symbol in regular long division).
_______________________
4x - 2 | 12x^2 + 2x - 9
2. Divide the first term of the dividend (12x^2) by the first term of the divisor (4x). Place the result, 3x, above the long division symbol.
3x
_______________________
4x - 2 | 12x^2 + 2x - 9
3. Multiply the divisor (4x - 2) by the result obtained in step 2 (3x). Write the product (12x^2 - 6x) below the dividend, aligned under the corresponding terms.
3x
_______________________
4x - 2 | 12x^2 + 2x - 9
- (12x^2 - 6x)
4. Subtract the product obtained in step 3 from the corresponding terms of the dividend. Write the result below.
3x
_______________________
4x - 2 | 12x^2 + 2x - 9
- (12x^2 - 6x)
_________________
8x - 9
5. Bring down the next term of the dividend (-9) and place it next to the result from step 4. This becomes the new dividend.
3x
_______________________
4x - 2 | 12x^2 + 2x - 9
- (12x^2 - 6x)
_________________
8x - 9
+ (- 9)
6. Divide the first term of the new dividend (8x) by the first term of the divisor (4x). Place the result, 2, above the long division symbol.
3x + 2
_______________________
4x - 2 | 12x^2 + 2x - 9
- (12x^2 - 6x)
_________________
8x - 9
+ (- 9)
__________________
8x - 11
7. Multiply the divisor (4x - 2) by the result obtained in step 6 (2). Write the product (8x - 4) below the new dividend, aligned under the corresponding terms.
3x + 2
_______________________
4x - 2 | 12x^2 + 2x - 9
- (12x^2 - 6x)
_________________
8x - 9
+ (- 9)
__________________
8x - 11
- (8x - 4)
8. Subtract the product obtained in step 7 from the corresponding terms of the new dividend. Write the result below.
3x + 2
_______________________
4x - 2 | 12x^2 + 2x - 9
- (12x^2 - 6x)
_________________
8x - 9
+ (- 9)
__________________
8x - 11
- (8x - 4)
__________________
- 5
9. The result of the division is the quotient (3x + 2) and the remainder (-5). Therefore, the polynomial 12x^2 + 2x - 9 can be expressed as the product of the divisor (4x - 2) and the quotient (3x + 2), plus the remainder (-5):
12x^2 + 2x - 9 = (4x - 2)(3x + 2) - 5
So, the correct answer is 1. 12x^2 + 2x - 9 = (4x - 2)(3x + 2) - 5.