The quotient of (x^4 + 5x^3 - 3x - 15) and (x^3 - 3) is a polynomial. What is the quotient?

Step-by-step solution

1. 1

   Compare degrees before dividing. The dividend has degree $4$ and the divisor degree $3$, so the quotient must have degree $4-3=1$. Any answer of degree $7$ (which would come from multiplying instead of dividing) can be ruled out immediately.

2. 2

   Divide the leading terms. $\frac{x^4}{x^3}=x$, so the first quotient term is $x$. Multiplying back, $x(x^3-3)=x^4-3x$.

3. 3

   Subtract and bring down. $(x^4+5x^3+0x^2-3x-15)-(x^4-3x)=5x^3+0x^2+0x-15$. The two $-3x$ terms cancel exactly, which is the hint that the division will come out even.

4. 4

   Divide again. $\frac{5x^3}{x^3}=5$, and $5(x^3-3)=5x^3-15$. Subtracting leaves $0$, so the remainder is zero and the quotient is $x+5$.

5. 5

   Confirm by factoring by grouping. $x^4+5x^3-3x-15=x^3(x+5)-3(x+5)=(x+5)(x^3-3)$. Cancelling the common factor $x^3-3$ gives the quotient $x+5$ directly, which matches the long division.

6. 6

   Numerical check at x = 2. The dividend is $16+40-6-15=35$ and the divisor is $8-3=5$; $35\div 5=7$, and the quotient $x+5$ at $x=2$ is also $7$.