What polynomial could you subtract from 5x^3 - 12x - x^2 + 9 - 12x^5 - 6x^4 to give a difference of 19 + 8x^3 - 18x - 19x^5 - 2x^2 - 8x^4?
Is the answer k (the polynomial) = 7x^5 + 2x^4 - 3x^3 + x^2 + 6x - 28?
To find the polynomial that you subtract from the expression on the left-hand side to give the given difference on the right-hand side, we can simplify both expressions and then compare the coefficients of each term.
First, let's rearrange the expressions:
5x^3 - 12x - x^2 + 9 - 12x^5 - 6x^4 - (19 + 8x^3 - 18x - 19x^5 - 2x^2 - 8x^4)
Simplifying both sides:
5x^3 - 12x - x^2 + 9 - 12x^5 - 6x^4 - 19 - 8x^3 + 18x + 19x^5 + 2x^2 + 8x^4
Combining like terms:
(5x^3 - 8x^3) + (18x - 12x) + (-1x^2 + 2x^2) + (-6x^4 + 8x^4) + (-12x^5 + 19x^5) + (9 - 19)
Simplifying further:
-3x^3 + 6x - x^2 + 2x^4 + 7x^5 - 10
Comparing the coefficients of each term, we can see that the answer should be:
k = 7x^5 + 2x^4 - 3x^3 - x^2 + 6x - 10
So, the answer you provided (k = 7x^5 + 2x^4 - 3x^3 + x^2 + 6x - 28) is not correct. The correct answer is:
k = 7x^5 + 2x^4 - 3x^3 - x^2 + 6x - 10
To verify if the polynomial k = 7x^5 + 2x^4 - 3x^3 + x^2 + 6x - 28 is the correct answer, we can subtract it from the original polynomial 5x^3 - 12x - x^2 + 9 - 12x^5 - 6x^4 and check if it equals the given difference 19 + 8x^3 - 18x - 19x^5 - 2x^2 - 8x^4.
Let's subtract k from the original polynomial:
(5x^3 - 12x - x^2 + 9 - 12x^5 - 6x^4) - (7x^5 + 2x^4 - 3x^3 + x^2 + 6x - 28)
First, let's group like terms:
-12x^5 + 7x^5 = -5x^5
-6x^4 - 2x^4 = -8x^4
5x^3 + 3x^3 = 8x^3
-x^2 - x^2 = -2x^2
-12x + 6x = -6x
9 + 28 = 37
37 - 19 = 18
Now, let's write the resulting polynomial:
-5x^5 - 8x^4 + 8x^3 - 2x^2 - 6x + 37
Finally, we can compare the resulting polynomial with the given difference: 19 + 8x^3 - 18x - 19x^5 - 2x^2 - 8x^4
We can see that the resulting polynomial is not equal to the given difference. Therefore, the answer k = 7x^5 + 2x^4 - 3x^3 + x^2 + 6x - 28 is incorrect.