Problem 38 - Algebra A Combined Function

Question

Factor out the GCF from each polynomial. $$ 12 x^{3}+16 x^{2}-8 x $$

Step-by-Step Solution

Answer

The GCF is 4x, and the polynomial factors to \(4x(3x^2 + 4x - 2)\).

Step 1: Identify the Greatest Common Factor (GCF)

The first step is to identify the GCF of the coefficients and the common variable factors in the polynomial. The coefficients are 12, 16, and -8. The GCF of 12, 16, and 8 is 4. The variable part of each term is \(x\), \(x^2\), and \(x^3\), so the common variable factor is \(x\). Thus, the GCF of the entire expression is \(4x\).

Step 2: Factor out the GCF from each term

Now, divide each term in the polynomial \(12x^3\), \(16x^2\), and \(-8x\) by the GCF \(4x\) and express the polynomial in factored form. \[12x^3 \div 4x = 3x^2\] \[16x^2 \div 4x = 4x\] \[-8x \div 4x = -2\] This results in the factored expression \(4x(3x^2 + 4x - 2)\).

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) plays a crucial role in simplifying polynomials. It's a bit like finding a common thread that links all terms together. Start by looking at the coefficients of the terms in the polynomial. For the polynomial:

• The coefficients are 12, 16, and -8.
• To find the GCF of these coefficients, consider the largest number that divides each of them. Here, it is 4.

Besides the coefficients, we also consider the variable part. In our polynomial:

• The variables are presented in forms of powers: \(x^3\), \(x^2\), and \(x\).
• The common variable factor in these terms is \(x\), precisely the lowest power of the variable.

Thus, the GCF of the polynomial \(12x^{3} + 16x^{2} - 8x\) is found by combining the GCF of the coefficients, 4, with the common variable factor, \(x\). Altogether, it provides us with a GCF of \(4x\). Breaking down the polynomial using the GCF simplifies further operations and is a key strategy in polynomial factorization.

Factoring Techniques

Factoring polynomials is like unraveling a complex puzzle. Techniques vary but generally involve identifying common elements and simplifying the expression. For our polynomial, once we have the GCF \(4x\), we proceed as follows:

• Take each term of the polynomial and divide it by the GCF.
• This involves calculating \(12x^3 \div 4x\), \(16x^2 \div 4x\), and \(-8x \div 4x\).

The results:

• \(\frac{12x^3}{4x} = 3x^2\)
• \(\frac{16x^2}{4x} = 4x\)
• \(\frac{-8x}{4x} = -2\)

This method of "factoring out" streamlines the expression into a simplified form: \(4x(3x^2 + 4x - 2)\). By using this technique, we've organized the polynomial into parts that are easier to understand and work with. Factoring out the GCF as an initial step is efficient, especially in larger algebraic operations.

Polynomial Division

Polynomial division is a technique used in factoring that involves dividing each term by a common factor—in this case, our GCF. This method is akin to traditional division but is applied to algebraic expressions instead of numbers. For the polynomial \(12x^3 + 16x^2 - 8x\), once the GCF is determined as \(4x\):

• We divide each term in the polynomial by \(4x\).
• This helps reduce the complexity of the polynomial and allows us to express it in a more manageable form.

After performing polynomial division:

• The quotient for \(12x^3 \div 4x\) is \(3x^2\).
• The quotient for \(16x^2 \div 4x\) is \(4x\).
• The quotient for \(-8x \div 4x\) is \(-2\).

Thus, the division allows us to neatly factor the original polynomial into \(4x(3x^2 + 4x - 2)\). Understanding this process equips you with a powerful tool for simplifying and solving polynomial equations efficiently. It acts as a bridge between recognizing a common element and transforming the polynomial into an easily usable form.