Answer :
Certainly! Let's solve the problem using synthetic division to find [tex]\(\left(x^4 - 16 x^2\right) \div (x - 4)\)[/tex].
### Step-by-Step Solution:
1. Set Up the Polynomial Coefficients:
The polynomial [tex]\(x^4 - 16 x^2\)[/tex] can be written in standard form as:
[tex]\[ x^4 + 0x^3 - 16x^2 + 0x + 0 \][/tex]
This gives us the coefficients: [tex]\([1, 0, -16, 0, 0]\)[/tex].
2. Set Up Synthetic Division:
Since we are dividing by [tex]\(x - 4\)[/tex], the root value we use is 4. This will be the number we use in our synthetic division setup.
3. Perform Synthetic Division:
[tex]\[ \begin{array}{r|rrrrr} \phantom{0}4 & 1 & 0 & -16 & 0 & 0 \\ \hline & 1 & 4 & 16 & 64 & 256 \\ \end{array} \][/tex]
- Write down the first coefficient (1) from the polynomial. This is our initial value.
- Multiply this number by the root (4) and write the result underneath the next coefficient.
- Add the column to get the new coefficient entry.
- Repeat the multiply and add process for each subsequent coefficient.
Following the procedure:
- Start with the first coefficient: 1
- Multiply by 4: [tex]\(1 \times 4 = 4\)[/tex]
- Add to the next coefficient (0): [tex]\(0 + 4 = 4\)[/tex]
- New coefficients array: [tex]\([1, 4]\)[/tex]
- Next number is 4:
- Multiply by 4: [tex]\(4 \times 4 = 16\)[/tex]
- Add to the next coefficient (-16): [tex]\(-16 + 16 = 0\)[/tex]
- New coefficients array: [tex]\([1, 4, 0]\)[/tex]
- Next number is 0:
- Multiply by 4: [tex]\(0 \times 4 = 0\)[/tex]
- Add to the next coefficient (0): [tex]\(0 + 0 = 0\)[/tex]
- New coefficients array: [tex]\([1, 4, 0, 0]\)[/tex]
- Last number is 0:
- Multiply by 4: [tex]\(0 \times 4 = 0\)[/tex]
- Add to the last coefficient (0): [tex]\(0 + 0 = 0\)[/tex]
- New coefficients array: [tex]\([1, 4, 0, 0, 0]\)[/tex]
The synthetic division table confirms our quotient, and [tex]\(0\)[/tex] as the remainder.
4. Interpret the Result:
The coefficients [tex]\([1, 4, 0, 0]\)[/tex] correspond to the polynomial [tex]\(x^3 + 4x^2 + 0x + 0\)[/tex], or simply [tex]\(x^3 + 4x^2\)[/tex] (ignoring the zero coefficients).
Therefore, the quotient of [tex]\(\left(x^4 - 16 x^2\right) \div (x - 4)\)[/tex] is:
[tex]\[ \boxed{x^3 + 4x^2} \][/tex]
The correct answer from the provided options is:
A. [tex]\(x^3 + 4 x^2\)[/tex]
### Step-by-Step Solution:
1. Set Up the Polynomial Coefficients:
The polynomial [tex]\(x^4 - 16 x^2\)[/tex] can be written in standard form as:
[tex]\[ x^4 + 0x^3 - 16x^2 + 0x + 0 \][/tex]
This gives us the coefficients: [tex]\([1, 0, -16, 0, 0]\)[/tex].
2. Set Up Synthetic Division:
Since we are dividing by [tex]\(x - 4\)[/tex], the root value we use is 4. This will be the number we use in our synthetic division setup.
3. Perform Synthetic Division:
[tex]\[ \begin{array}{r|rrrrr} \phantom{0}4 & 1 & 0 & -16 & 0 & 0 \\ \hline & 1 & 4 & 16 & 64 & 256 \\ \end{array} \][/tex]
- Write down the first coefficient (1) from the polynomial. This is our initial value.
- Multiply this number by the root (4) and write the result underneath the next coefficient.
- Add the column to get the new coefficient entry.
- Repeat the multiply and add process for each subsequent coefficient.
Following the procedure:
- Start with the first coefficient: 1
- Multiply by 4: [tex]\(1 \times 4 = 4\)[/tex]
- Add to the next coefficient (0): [tex]\(0 + 4 = 4\)[/tex]
- New coefficients array: [tex]\([1, 4]\)[/tex]
- Next number is 4:
- Multiply by 4: [tex]\(4 \times 4 = 16\)[/tex]
- Add to the next coefficient (-16): [tex]\(-16 + 16 = 0\)[/tex]
- New coefficients array: [tex]\([1, 4, 0]\)[/tex]
- Next number is 0:
- Multiply by 4: [tex]\(0 \times 4 = 0\)[/tex]
- Add to the next coefficient (0): [tex]\(0 + 0 = 0\)[/tex]
- New coefficients array: [tex]\([1, 4, 0, 0]\)[/tex]
- Last number is 0:
- Multiply by 4: [tex]\(0 \times 4 = 0\)[/tex]
- Add to the last coefficient (0): [tex]\(0 + 0 = 0\)[/tex]
- New coefficients array: [tex]\([1, 4, 0, 0, 0]\)[/tex]
The synthetic division table confirms our quotient, and [tex]\(0\)[/tex] as the remainder.
4. Interpret the Result:
The coefficients [tex]\([1, 4, 0, 0]\)[/tex] correspond to the polynomial [tex]\(x^3 + 4x^2 + 0x + 0\)[/tex], or simply [tex]\(x^3 + 4x^2\)[/tex] (ignoring the zero coefficients).
Therefore, the quotient of [tex]\(\left(x^4 - 16 x^2\right) \div (x - 4)\)[/tex] is:
[tex]\[ \boxed{x^3 + 4x^2} \][/tex]
The correct answer from the provided options is:
A. [tex]\(x^3 + 4 x^2\)[/tex]
