Using synthetic division, find the quotient of x3−13x−12

Question

𝑥
3
−
13
𝑥
−
12
and x−4
𝑥
−
4
.(1 point)
Responses

x2−9x−48
𝑥
2
−
9
𝑥
−
48
x squared minus 9 x minus 48

x2+4x+3 +12x−4
𝑥
2
+
4
𝑥
+
3

+
12
𝑥
−
4
x2+4x+3 +12x−4
𝑥
2
+
4
𝑥
+
3

+
12
𝑥
−
4

x2−4x+3−24x−4
𝑥
2
−
4
𝑥
+
3
−
24
𝑥
−
4
x squared minus 4 x plus 3 minus the fraction with numerator 24 and denominator x minus 4

x2+4x+3

Answer 1

To use synthetic division to divide the polynomial \( x^3 - 13x - 12 \) by \( x - 4 \), we will perform the following steps:

1. Write down the coefficients of the polynomial. The polynomial \( x^3 - 13x - 12 \) can be rewritten as:

   • Coefficient of \( x^3 \): 1
   • Coefficient of \( x^2 \): 0 (since there is no \( x^2 \) term)
   • Coefficient of \( x^1 \): -13
   • Constant term: -12

   Thus, the coefficients are \( [1, 0, -13, -12] \).

2. Use the root of the divisor \( x - 4 \), which is \( 4 \), for synthetic division. We set up synthetic division as follows:

4 |  1   0   -13   -12
   |      4    16   12
   -----------------------
      1   4    3     0

3. Carry down the first coefficient, then multiply it by \( 4 \) and write the result under the next coefficient. We continue this process as shown above:

   • First, bring down the \( 1 \).
   • Multiply \( 1 \) by \( 4 \) to get \( 4 \).
   • Add \( 0 + 4 = 4 \).
   • Multiply \( 4 \) by \( 4 \) to get \( 16 \).
   • Add \( -13 + 16 = 3 \).
   • Multiply \( 3 \) by \( 4 \) to get \( 12 \).
   • Add \( -12 + 12 = 0 \) (we find a remainder of \( 0 \)).

4. The coefficients of the quotient polynomial are the numbers at the bottom row (excluding the last one). They represent the polynomial \( x^2 + 4x + 3 \).

Hence, the final result, which is the quotient when \( x^3 - 13x - 12 \) is divided by \( x - 4 \), is:

\[ \boxed{x^2 + 4x + 3} \]