Find each product. See Example 5. (x + 1) (x + 1) (x - 1) (x - 1)

Verified step by step guidance

1. Recognize that the expression is a product of two pairs of binomials: \((x + 1)(x + 1)\) and \((x - 1)(x - 1)\).
2. Rewrite each pair as a square: \((x + 1)^2\) and \((x - 1)^2\).
3. Recall the formula for the square of a binomial: \((a + b)^2 = a^2 + 2ab + b^2\) and \((a - b)^2 = a^2 - 2ab + b^2\).
4. Expand each square using the formula: \((x + 1)^2 = x^2 + 2x + 1\) and \((x - 1)^2 = x^2 - 2x + 1\).
5. Multiply the two expanded expressions: \((x^2 + 2x + 1)(x^2 - 2x + 1)\), and then use the distributive property (FOIL) to find the product.

Key Concepts

Polynomial Multiplication

Polynomial multiplication involves multiplying two or more polynomial expressions by applying the distributive property. Each term in one polynomial is multiplied by every term in the other, and like terms are combined to simplify the result.

Difference of Squares

The difference of squares is a special product formula: (a + b)(a - b) = a² - b². Recognizing this pattern helps simplify expressions quickly without full expansion, especially when multiplying conjugate binomials.

Combining Like Terms

After multiplying polynomials, like terms—terms with the same variable raised to the same power—must be combined by adding or subtracting their coefficients. This step simplifies the expression to its standard polynomial form.