Question

Completely factor the expression.\(x^{3}-4 x^{2}\)

Short answer

The completely factored form of the expression \(x^{3}-4 x^{2}\) is \(x^{2}(x-4)\).

Step-by-step solution

Identify the Common Factor

In the given expression, \(x^{3}-4 x^{2}\), the common factor between the two terms is \(x^{2}\).

Factor out the Common Factor

Once the common factor is identified, the next step is to factor out this from the original expression. Factor out \(x^{2}\) from \(x^{3}-4 x^{2}\) to get \(x^{2}(x-4)\).

Check the Result

Make sure that the factored form of the expression is correct. If you expand \(x^{2}(x-4)\) using the distributive law, you should get back to the original expression, \(x^{3}-4 x^{2}\).

Key concepts

Common Factor

When working with polynomial expressions, a common factor is an element present in each term that can be factored out to simplify the expression. Think of it as the number or variable that divides each term evenly. In the expression \(x^3 - 4x^2\), the common factor is identified by looking for the greatest exponent of \(x\) that appears in each term. Here, both terms share \(x^2\) as a common factor:- The first term, \(x^3\), can be divided by \(x^2\) resulting in \(x\).- The second term, \(-4x^2\), divided by \(x^2\), results in \(-4\).Recognizing and factoring out the common factor simplifies polynomial expressions and is a crucial first step in polynomial factorization.

Polynomial Expression

A polynomial expression consists of terms made up of variables raised to exponents and multiplied by coefficients. Each term in a polynomial is separated by addition or subtraction.The polynomial \(x^3 - 4x^2\) has two terms:- The first term is \(x^3\), which includes a variable \(x\) raised to the power of three.- The second term is \(-4x^2\), featuring a coefficient of \(-4\) and the variable \(x\) raised to the power of two.Understanding the structure of polynomial expressions helps in identifying common factors and simplifies the process of factoring and expanding expressions. Always remember, the degree of the polynomial is determined by the highest power in the expression.

Distributive Law

The distributive law, a fundamental algebraic principle, allows us to simplify expressions by distributing a factor over terms within parentheses. It states: for any numbers or variables \(a\), \(b\), and \(c\), the expression \(a(b + c)\) equals \(ab + ac\).By applying the distributive law, one can verify the correctness of factorization. In the exercise, the expression is factored as \(x^2(x-4)\).- To expand \(x^2(x-4)\): - Distribute \(x^2\) across \(x - 4\). - Resulting in \(x^2 \times x - x^2 \times 4\). - This simplifies to \(x^3 - 4x^2\).Checking the result by expansion ensures that the factorization is accurate, allowing the expression to revert back to its original form. The distributive law is a powerful tool in both simplifying and expanding polynomial expressions.