https://file.aiquickdraw.com/mathbot/file/1727022411706zxf1dtgl.jpg

Walk through this problem using synthetic division of 1

The polynomial given in the image is:

\[
x^4 - 4x^3 + 7x^2 - 16x + 12
\]

You want to perform synthetic division on this polynomial using 1 as the divisor. Let's go through the steps of synthetic division.

### Step 1: Set up the synthetic division
To begin, we write down the coefficients of the polynomial:

\[
1, -4, 7, -16, 12
\]

The divisor is \( x - 1 \), so we use \( 1 \) for synthetic division.

### Step 2: Perform the synthetic division

We set it up as follows:

\[
\begin{array}{r|rrrrr}
1 & 1 & -4 & 7 & -16 & 12 \\
  &   & 1  & -3 & 4  & -12 \\
\hline
  & 1 & -3 & 4  & -12 & 0 \\
\end{array}
\]

### Step 3: Explanation of each step

1. **Bring down** the leading coefficient (1).
2. **Multiply** this number by the divisor (1), and write the result under the next coefficient: \( 1 \times 1 = 1 \).
3. **Add** the numbers in the second column: \( -4 + 1 = -3 \).
4. **Multiply** this result by the divisor (1), and write it under the next coefficient: \( -3 \times 1 = -3 \).
5. **Add** the numbers in the third column: \( 7 + (-3) = 4 \).
6. **Multiply** this result by the divisor (1): \( 4 \times 1 = 4 \), and add to the next column: \( -16 + 4 = -12 \).
7. **Multiply** the result by the divisor (1): \( -12 \times 1 = -12 \), and add to the last column: \( 12 + (-12) = 0 \).

### Step 4: Conclusion

The quotient from this synthetic division is:

\[
x^3 - 3x^2 + 4x - 12
\]

And the remainder is 0, indicating that \( x - 1 \) is a factor of the polynomial.

Would you like more details, or do you have any questions about synthetic division?

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Here are 5 related questions you can explore next:

1. What is the factorization of the quotient \( x^3 - 3x^2 + 4x - 12 \)?
2. How can you confirm that \( x = 1 \) is a root of the polynomial using substitution?
3. What happens if you perform synthetic division using a different root, such as \( x = 2 \)?
4. How does synthetic division differ from long division in polynomials?
5. How can you use the Remainder Theorem to quickly verify the remainder from synthetic division?

**Tip:** Synthetic division is a shortcut method, but it only works when dividing by linear factors of the form \( x - c \).

This step-by-step guide walks through the synthetic division of the polynomial x^4 - 4x^3 + 7x^2 - 16x + 12 using the divisor x - 1. Learn how to divide polynomials, understand the quotient and remainder, and explore the applications of the Remainder and Factor Theorems. Suitable for students in Grades 10-12.

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