Problem 100 - Introductory Algebra for College Students

Question

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$3 x z^{2}-72 x z+432 x$$

Step-by-Step Solution

Answer

The completely factorised form of the polynomial \(3xz^2 - 72xz + 432x\) is \(3x(z - 12)^2\).

Step 1: Identify the common factor

Find the greatest common factor of the three terms, which is \(3x\).

Step 2: Factor the polynomial

Write the polynomial as the product of the common factor and another expression: \(3xz^2 - 72xz + 432x = 3x(z^2 - 24z + 144)\). The expression in the parentheses is a quadratic trinomial that can be factored further.

Step 3: Factor the quadratic trinomial

Factorize the quadratic trinomial in the parentheses: \(z^2 - 24z + 144 = (z - 12)^2\). So the completely factorised polynomial is: \(3x(z - 12)^2\).

Step 4: Check the solution

Multiply the factors to check the solution: \(3x(z - 12)^2 = 3xz^2 - 72xz + 432x\), which is the original polynomial, so the factorization is correct.

Key Concepts

Polynomials

Polynomials are mathematical expressions that involve a sum of powers of variables with coefficients. Each term in a polynomial includes a coefficient (which is a number), one or more variables, and each variable has a non-negative integer exponent. For example, in the expression \(3xz^2-72xz+432x\), each term is part of a polynomial with the variable \(z\) and coefficients \(3x, -72x,\) and \(432x\).

Polynomials can contain terms with several variables, making them multivariable polynomials. In this particular expression, we see the presence of two variables, \(x\) and \(z\). Handling multivariable polynomials often involves looking for common factors and organizing these expressions in a much simpler form.

Understanding polynomials helps in various algebraic processes, such as factoring, expanding, finding roots, and graphing, being foundational for algebra and calculus.

Factoring

Factoring is the process of breaking down a complex expression into simpler 'factors' that, when multiplied together, will produce the original expression. This is an essential skill in algebra as it simplifies solving equations and further operations. To begin factoring, identify common elements across all terms.

In the expression \(3xz^2 - 72xz + 432x\), we identify \(3x\) as the greatest common factor for all terms. By pulling out this factor, the expression simplifies to \(3x(z^2 - 24z + 144)\), making it more manageable. Factoring can often be done through various methods such as grouping, substitution, or by looking for patterns like difference of squares and sum/difference of cubes.

The simplified form opens up opportunities to factor further, especially if any trinomial (three-term expression) in the polynomial can still be broken down.

Quadratic Trinomial

A quadratic trinomial is a polynomial with three terms where the highest power of the variable is 2 (hence, 'quadratic'). The standard form of a quadratic trinomial is given by \(ax^2 + bx + c\).

In our example, the expression \(z^2 - 24z + 144\) inside the parentheses fits this form, where \(a=1\), \(b=-24\), and \(c=144\). This trinomial can often be factored into the product of two binomials. If it is a perfect square trinomial, it takes the form \((z-p)^2\), where \(p\) is a number that satisfies the relationship \(p=-b/2\).

In the given expression, evaluating the perfect square pattern shows that \(z^2 - 24z + 144 = (z-12)^2\). It’s always helpful to verify factorization by expanding back to ensure it returns the initial expression. This falls under the broader skill of recognizing patterns in algebra, critical for solving more complex equations and problems effectively.