Question

Simplify \((x-2)^2\)

Short answer

The simplified form of \((x-2)^2\) is \(x^{2} - 4x + 4\).

Step-by-step solution

Identify a and b

In the expression \((x-2)^2\), identify \(a\) and \(b\). Here, \(a = x\) and \(b = 2\).

Substitute a and b in the formula

Substitute \(a = x\) and \(b = 2\) in the formula \(a^{2} - 2ab + b^{2}\) . This gives \(x^{2} - 2*(x*2) + 2^{2}\).

Simplify the expression

Simplify \(x^{2} - 2x*2 + 2^{2}\). This leads to \(x^{2} - 4x + 4\).

Key concepts

Polynomial Expansion

Polynomial expansion refers to the process of expanding an expression raised to a power, such as \[(x-2)^2\]. In this context, it involves the distribution and combination of terms. This step is crucial for simplifying complex algebraic expressions, allowing us to work with them more easily.

When expanding polynomials, it’s important to multiply each term consistently. For \((x-2)^2\), it would help to write down the expression as \((x-2) \times (x-2)\). Then, use the distributive property (often called the FOIL method for binomials) to expand it:

• First: Multiply the first terms in each binomial: \(x \times x = x^2\).


• Outside: Multiply the outer terms: \(x \times (-2) = -2x\).


• Inside: Multiply the inner terms: \(-2 \times x = -2x\).


• Last: Multiply the last terms: \((-2) \times (-2) = 4\).

Finally, combine like terms: \(x^2 - 2x - 2x + 4 = x^2 - 4x + 4\). This expansion method ties closely with the Binomial Theorem, which provides a quicker method for certain kinds of expressions.

Binomial Theorem

Binomial Theorem is a fantastic tool that makes polynomial expansion easier, specifically for expressions raised to a power.

It gives us a straightforward way to expand a binomial expression without going through each individual multiplication, important when dealing with higher powers.

The theorem states that:\[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]This formula provides a way to find each term for the expansion's result, where \(\binom{n}{k}\) is a binomial coefficient, \(a\) and \(b\) are components of the binomial, and \(n\) is the power of the binomial. For smaller powers, applying the step-by-step method rather than the theorem can often feel more intuitive, but understanding the Binomial Theorem prepares you for more complex tasks.

Applying this to \((x-2)^2\) shows us why the solution \(x^2 - 4x + 4\) follows directly from both the theorem and simple multiplication.

Algebraic Expressions

Algebraic expressions are the heart of simplifying tasks in algebra. These are combinations of numbers, variables, and operations. They can become very complex, but breaking them down into understandable parts is key.

An expression such as \((x-2)^2\) involves variables \((x)\) and constants \((2)\). Each of the terms in an expanded algebraic expression plays a unique role.

For \(x^2 - 4x + 4\), each of these:

• \(x^2\) is the quadratic term, gives the curve of a graph its parabolic shape.
• \(-4x\) is the linear term, represents a straight-line slope.
• \(4\) is the constant term, shifts the graph up or down on the coordinate plane.

Understanding each part’s role in the equation helps us grasp how expressions behave. Simplifying expressions relies on this comprehension, hence why step-by-step simplification (like substituting in the polynomial expansion formula) strengthens our foundation in algebra.