Using polynomial long division to divide.
25x^3+40x^2-25x+10
by
-5x^2-3x-2
To divide the polynomial 25x^3 + 40x^2 - 25x + 10 by -5x^2 - 3x - 2 using polynomial long division, follow these steps:
Step 1: Arrange the terms in descending order of degrees and ensure that all degrees are represented.
The dividend: 25x^3 + 40x^2 - 25x + 10
The divisor: -5x^2 - 3x - 2
Step 2: Divide the highest degree term of the dividend by the highest degree term of the divisor.
Dividing 25x^3 by -5x^2, we get -5x.
Step 3: Multiply the divisor by the result from Step 2.
Multiplying -5x by -5x^2 - 3x - 2, we get -5x(-5x^2 - 3x - 2) = 25x^3 + 15x^2 + 10x.
Step 4: Subtract the result obtained in Step 3 from the dividend.
Subtracting 25x^3 + 15x^2 + 10x from 25x^3 + 40x^2 - 25x + 10, we get (40x^2 - 15x - 10x) + 10.
Simplifying, we obtain 40x^2 - 25x - 10.
Step 5: Repeat Steps 2-4 with the new reduced dividend.
Dividing 40x^2 by -5x^2, we get -8.
Multiplying -8 by -5x^2 - 3x - 2, we get -8(-5x^2 - 3x - 2) = 40x^2 + 24x + 16.
Subtracting 40x^2 + 24x + 16 from 40x^2 - 25x - 10, we get (-25x - 24x) - 10 + 16.
Simplifying, we obtain -49x + 6.
Step 6: Repeat Steps 2-4 with the new reduced dividend.
Dividing -49x by -5x^2, we get 9.8x.
Multiplying 9.8x by -5x^2 - 3x - 2, we get 9.8x(-5x^2 - 3x - 2) = -49x^3 - 29.4x^2 - 19.6x.
Subtracting -49x^3 - 29.4x^2 - 19.6x from -49x + 6, we get (29.4x^2 + 0.4x) + 6.
Simplifying, we obtain 29.4x^2 + 0.4x + 6.
Step 7: Since the degree of the reduced dividend (29.4x^2 + 0.4x + 6) is less than the divisor, the division process is complete.
Therefore, the quotient is -5x - 8 + 9.8x, which simplifies to 4.8x - 8, and the remainder is 29.4x^2 + 0.4x + 6.
Hence, the division of 25x^3 + 40x^2 - 25x + 10 by -5x^2 - 3x - 2 using polynomial long division results in the quotient of 4.8x - 8 and the remainder of 29.4x^2 + 0.4x + 6.