Final answer:
The quotient of (x⁵-3x³ -3x²-10x+15) and (x^(2)-5) is x³-3x-25 with a remainder of -110.
Step-by-step explanation:
To find the quotient of (x⁵-3x³ -3x²-10x+15) and (x^(2)-5), we will use polynomial long division.
Step 1: Start by dividing the leading terms: x⁵ ÷ x² = x³.
Step 2: Multiply the divisor (x²-5) by the quotient (x³) and subtract the result from the dividend (x⁵-3x³): (x³)(x²-5) = x⁵-5x³. Subtracting this from the dividend gives -3x³-10x.
Step 3: Bring down the next term: -3x³-10x+15.
Step 4: Divide the new leading term (-3x³) by the leading term of the divisor (x²): -3x³ ÷ x² = -3x.
Step 5: Multiply the divisor (x²-5) by the new quotient (-3x) and subtract the result from the previous step: (-3x)(x²-5) = -3x³+15x. Subtracting this from the previous step gives -25x+15.
Step 6: Bring down the next term (-25x+15).
Step 7: Divide the new leading term (-25x) by the leading term of the divisor (x²): -25x ÷ x² = -25.
Step 8: Multiply the divisor (x²-5) by the new quotient (-25) and subtract the result from the previous step: (-25)(x²-5) = -25x²+125. Subtracting this from the previous step gives -110.
Step 9: Since there are no more terms to bring down, the quotient is x³-3x-25 with a remainder of -110.