- Start with the identity \((x+1)^3 = (x+1)(x^2+2x+1)\).
- Apply the distributive property to multiply each term: \(x(x^2+2x+1) + 1(x^2+2x+1) = x^3+2x^2+x + x^2+2x+1\).
- Combine like terms to arrive at the expanded form: \(x^3 + 3x^2 + 3x + 1\).
- The final polynomial is \(\boxed{x^3 + 3x^2 + 3x + 1}\).
Explanation
- Analysis of the Problem
To expand the expression \((x+1)^3\), we are given the relationship \((x+1)^3 = (x+1)(x+1)^2\). We know that the expansion of \((x+1)^2\) is \(x^2 + 2x + 1\). Our goal is to perform the multiplication \((x+1)(x^2 + 2x + 1)\) and identify the resulting coefficients in the polynomial form \(x^3 + \square x^2 + \square x + \square\).
- Expansion and Simplification
To multiply \((x+1)\) by \((x^2 + 2x + 1)\), we use the distributive property. We distribute each term in the first binomial across all terms in the trinomial:
\[(x+1)(x^2 + 2x + 1) = x(x^2 + 2x + 1) + 1(x^2 + 2x + 1)\]
Now, we perform the multiplication for each part:
\[x(x^2 + 2x + 1) = x^3 + 2x^2 + x\]
\[1(x^2 + 2x + 1) = x^2 + 2x + 1\]
Adding these two results together:
\[(x^3 + 2x^2 + x) + (x^2 + 2x + 1)\]
Finally, we combine like terms:
\[x^3 + (2x^2 + x^2) + (x + 2x) + 1 = x^3 + 3x^2 + 3x + 1\]
Comparing this to the target form \(x^3 + \square x^2 + \square x + \square\), we identify the missing coefficients as 3, 3, and 1.
Examples
Binomial expansion, like the one used for \((x+1)^3\), is fundamental in fields such as engineering, physics, and economics. For example, if a company's profit grows relative to a factor \(x\) such that the growth is modeled by a cubic function, knowing the exact polynomial coefficients allows them to predict future performance more accurately. By expanding expressions, we can easily calculate the sensitivity of these models to changes in input, making complex systems much more manageable and predictable.