https://file.aiquickdraw.com/mathbot/file/1726067688982c7yvm79v.jpg

The expression in the image is:

\[
x^3 - 8x^2 + 19x - 12 \div (x - 1)
\]

This represents the division of a cubic polynomial by a linear factor \((x - 1)\). To solve this, we will use polynomial long division or synthetic division to simplify the expression.

### Polynomial Long Division:
1. Divide the first term of the cubic polynomial \(x^3\) by the first term of the divisor \(x\), giving us \(x^2\).
2. Multiply \(x^2\) by \((x - 1)\), which gives \(x^3 - x^2\).
3. Subtract this from the original polynomial: \((x^3 - 8x^2)\) minus \((x^3 - x^2)\) gives \(-7x^2\).
4. Divide \(-7x^2\) by \(x\), giving \(-7x\).
5. Multiply \(-7x\) by \((x - 1)\), which gives \(-7x^2 + 7x\).
6. Subtract \((-7x^2 + 19x)\) minus \((-7x^2 + 7x)\), giving \(12x\).
7. Divide \(12x\) by \(x\), giving \(12\).
8. Multiply \(12\) by \((x - 1)\), which gives \(12x - 12\).
9. Subtract: \((12x - 12)\) minus \((12x - 12)\), which gives 0.

Thus, the quotient is:

\[
x^2 - 7x + 12
\]

Hence, the result of dividing \(x^3 - 8x^2 + 19x - 12\) by \((x - 1)\) is:

\[
x^2 - 7x + 12
\]

Do you need further clarification or a detailed breakdown of any step?

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Here are some related questions:

1. How do you perform synthetic division on polynomials?
2. What is the remainder theorem, and how does it relate to polynomial division?
3. Can the quotient \(x^2 - 7x + 12\) be factored further?
4. How do you solve cubic polynomials when factoring is not straightforward?
5. What are some applications of polynomial division in calculus?

**Tip**: When dividing polynomials, ensure that all powers of the variable are present by adding any missing terms with a coefficient of zero.

Divide the polynomial x^3 - 8x^2 + 19x - 12 by (x - 1)

This page walks through dividing the cubic polynomial x^3 - 8x^2 + 19x - 12 by (x - 1) using polynomial long division. The solution covers important algebraic techniques and provides a detailed explanation of each step. Suitable for students in Grades 10-12.

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