Problem 7 - Algebra 2 and Trigonometry

Question

In \(3-8,\) write each polynomial as the product of its greatest common monomial factor and a polynomial. $$ 4 a-12 a b+16 a^{2} $$

Step-by-Step Solution

Answer

The expression can be factored as \(4a(1 - 3b + 4a)\).

Step 1: Identify the Greatest Common Monomial Factor

Look at each term in the expression \(4a - 12ab + 16a^2\). Identify the common factors in each term. Each term contains the factor \(a\), and the coefficients 4, 12, and 16 share a common factor of 4. Thus, the greatest common monomial factor is \(4a\).

Step 2: Divide Each Term by the Greatest Common Monomial Factor

Divide every term in the polynomial by \(4a\) to find the remaining polynomial. \[ \begin{align*} \frac{4a}{4a} &= 1, \ \frac{-12ab}{4a} &= -3b, \ \frac{16a^2}{4a} &= 4a. \end{align*} \] So, the remaining polynomial is \(1 - 3b + 4a\).

Step 3: Write the Original Polynomial as the Product

Combine the greatest common monomial factor with the polynomial obtained in Step 2. The original polynomial \(4a - 12ab + 16a^2\) can be written as \(4a(1 - 3b + 4a)\).

Key Concepts

Greatest Common Monomial Factor

In algebra, one useful technique is identifying the **greatest common monomial factor** of a polynomial.The greatest common monomial factor is the largest factor that divides each term of a polynomial without leaving a remainder.This concept helps simplify expressions, making them easier to solve or manipulate.
To find the greatest common monomial factor:

• Look for common factors in the coefficients of each term.
• Identify common variables and their smallest power present in all terms.

For example, in the polynomial \(4a - 12ab + 16a^2\):

• The coefficients 4, 12, and 16 share the common factor 4.
• The variable \(a\) is present in every term.

Thus, the greatest common monomial factor is \(4a\). By using this factor, you can write the polynomial in a more simplified form.

Factoring Polynomials

Factoring polynomials means rewriting them as a product of simpler polynomials.This process helps in solving equations and understanding polynomial properties better.
For instance, if you want to factor the polynomial \(4a - 12ab + 16a^2\), you begin by extracting the greatest common monomial factor, which we've identified as \(4a\).After factoring out \(4a\), we divide each term in the polynomial by this factor:

• \(\frac{4a}{4a} = 1\)
• \(\frac{-12ab}{4a} = -3b\)
• \(\frac{16a^2}{4a} = 4a\)

By doing this, we obtain the polynomial \(1 - 3b + 4a\) which, when multiplied by the factor \(4a\), recreates the original polynomial.This approach shows how breaking down a polynomial into a product of simpler factors can be very advantageous.

Algebraic Expressions

Algebraic expressions are mathematical phrases containing numbers, variables, and operations.They are a fundamental element of algebra, serving as the building blocks for equations and functions.
An algebraic expression can range from a simple term like \(3x\) to more complex polynomials like \(4a - 12ab + 16a^2\).The terms in an expression are separated by addition or subtraction.
In the polynomial example, each part such as \(4a\), \(-12ab\), and \(16a^2\) is considered a term.Understanding the structure of an algebraic expression helps in operations like simplifying, evaluating, and particularly, factoring.Practicing with such expressions enhances both arithmetic skills and abstract thinking capabilities, crucial in more advanced mathematical studies.