(x^2-x+1)(x^2+x+1)

Jasonbradley

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Jun 17, 2024

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Expanding the Expression: (x^2 - x + 1)(x^2 + x + 1)

This expression involves multiplying two quadratic polynomials. We can expand it using the distributive property (also known as FOIL) or by recognizing a pattern.

Using the Distributive Property (FOIL)

FOIL stands for First, Outer, Inner, Last. It's a mnemonic device to remember all the terms that need to be multiplied when expanding two binomials.

1. First: Multiply the first terms of each binomial: x² * x² = x⁴
2. Outer: Multiply the outer terms of each binomial: x² * x = x³
3. Inner: Multiply the inner terms of each binomial: -x * x² = -x³
4. Last: Multiply the last terms of each binomial: -x * x = -x²

Now we need to multiply 1 from the first binomial by each term in the second binomial:

• 1 * x² = x²
• 1 * x = x
• 1 * 1 = 1

Combining all the terms:

x⁴ + x³ - x³ - x² + x² + x + 1

Simplifying by combining like terms:

x⁴ + x + 1

Using a Pattern Recognition

Notice that the two binomials are conjugates of each other, meaning they have the same terms but opposite signs in the middle. This leads to a pattern:

(a + b)(a - b) = a² - b²

In our case, a = x² and b = x - 1.

Applying the pattern:

(x² - x + 1)(x² + x + 1) = (x²)² - (x - 1)²

Expanding the square:

= x⁴ - (x² - 2x + 1)

Simplifying:

= x⁴ - x² + 2x - 1

= x⁴ + x + 1

Conclusion

Both methods lead to the same simplified expression: x⁴ + x + 1. Recognizing the pattern can make the expansion faster, but understanding the distributive property is crucial for more complex expressions.