Isiah divides 2x^3 - x^2 + 2x + 5 by x + 1 using a division table. The last quotient entry is labelled A. What is the value of A? Choices: -5, 5, 1x, 5x.

Step-by-step solution

1. 1

   Work out what the table is computing. A division table records the quotient one term at a time. Since the dividend has degree $3$ and the divisor degree $1$, the quotient has degree $2$ and therefore exactly three entries: an $x^2$ term, an $x$ term, and a constant. $A$ is the constant entry.

2. 2

   First quotient term. $\frac{2x^3}{x}=2x^2$. Multiplying back gives $2x^2(x+1)=2x^3+2x^2$, and subtracting leaves $-3x^2+2x+5$.

3. 3

   Second quotient term. $\frac{-3x^2}{x}=-3x$. Multiplying back gives $-3x(x+1)=-3x^2-3x$, and subtracting leaves $5x+5$.

4. 4

   Third quotient term, which is A. $\frac{5x}{x}=5$. Multiplying back gives $5(x+1)=5x+5$, and subtracting leaves a remainder of $0$. So $A=5$.

5. 5

   Rule out the distractors. $-5$ has the wrong sign, and $1x$ and $5x$ are of degree $1$; the last entry of a degree-$2$ quotient must be a constant, so only $5$ can be correct.

6. 6

   Check by multiplying out. $(x+1)(2x^2-3x+5)=2x^3-3x^2+5x+2x^2-3x+5=2x^3-x^2+2x+5$, which is the original dividend exactly.