A student divides x³ - 2x² - 10x + 21 by x² + x - 7 using a division table. The first quotient entry is x and the second entry is labeled A. What is the value of A?

Step-by-step solution

1. 1

   Understand what a division table entry is. A division table is long division rearranged into a grid: each quotient entry is chosen so that, when multiplied by the leading term of the divisor, it reproduces the current leading term of what is left of the dividend. So $A$ is nothing mysterious — it is the second term of the quotient.

2. 2

   Recover the first entry to confirm the setup. Divide the leading term of the dividend by the leading term of the divisor:

   $\frac{x^3}{x^2} = x$

   This matches the given first entry, so the table is being filled left to right in the usual way.

3. 3

   Multiply back and subtract. Multiply the whole divisor by that first entry and subtract the product from the dividend — this is the step that clears the $x^3$ term:

   $x(x^2 + x - 7) = x^3 + x^2 - 7x$

   $(x^3 - 2x^2 - 10x + 21) - (x^3 + x^2 - 7x) = -3x^2 - 3x + 21$

4. 4

   Repeat the leading-term division to get $A$. The new dividend leads with $-3x^2$, and the divisor still leads with $x^2$:

   $A = \frac{-3x^2}{x^2} = -3$

   It is a constant, not $-3x$, because the degrees cancel exactly.

5. 5

   Finish the table and check the remainder. Multiplying back gives $-3(x^2 + x - 7) = -3x^2 - 3x + 21$, which subtracts to $0$. So the division is exact:

   $x^3 - 2x^2 - 10x + 21 = (x^2 + x - 7)(x - 3)$

   Expanding the right side reproduces the dividend term for term, which confirms both the quotient $x - 3$ and the value $A = -3$.