In Exercises 23–34, find each product using either a horizontal or a vertical format. (x−1)(x²+x+1)

Verified step by step guidance

1. Start by distributing each term in the first polynomial, \((x - 1)\), to each term in the second polynomial, \((x^2 + x + 1)\).
2. First, distribute \(x\) from \((x - 1)\) to each term in \((x^2 + x + 1)\):
3. Calculate \(x \cdot x^2 = x^3\).
4. Calculate \(x \cdot x = x^2\).
5. Calculate \(x \cdot 1 = x\).

Key Concepts

Polynomial Multiplication

Polynomial multiplication involves distributing each term of one polynomial to every term of another polynomial. This process can be visualized using the distributive property, where each term in the first polynomial is multiplied by each term in the second. The results are then combined by adding like terms to simplify the expression.

Horizontal and Vertical Formats

Horizontal and vertical formats refer to different methods of organizing polynomial multiplication. The horizontal format lays out the polynomials in a single line, while the vertical format stacks them like traditional multiplication. Both methods ultimately yield the same result, but the choice of format can affect clarity and ease of calculation.

Combining Like Terms

Combining like terms is a crucial step in simplifying polynomial expressions after multiplication. Like terms are those that have the same variable raised to the same power. By adding or subtracting the coefficients of these terms, one can simplify the polynomial to its most concise form, making it easier to interpret and use in further calculations.