Problem 43 - Algebra A Combined Function

Question

Factor out the GCF from each polynomial. $$ 8 x^{5}+16 x^{4}-20 x^{3}+12 $$

Step-by-Step Solution

Answer

The factored form is \(4(2x^5 + 4x^4 - 5x^3 + 3)\).

Step 1: Identify the Greatest Common Factor (GCF)

Examine the coefficients and terms in the polynomial \(8x^5 + 16x^4 - 20x^3 + 12\). The coefficients are 8, 16, -20, and 12. The GCF of these numbers is 4.

Step 2: Factor the GCF from Each Term

Divide each term of the polynomial by the GCF, which is 4. This gives: \(\frac{8x^5}{4} + \frac{16x^4}{4} - \frac{20x^3}{4} + \frac{12}{4}\). Simplifying these terms results in \(2x^5 + 4x^4 - 5x^3 + 3\).

Step 3: Write the Factored Form

Now, express the polynomial with the GCF factored out: \(4(2x^5 + 4x^4 - 5x^3 + 3)\).

Key Concepts

Greatest Common Factor

In mathematics, the Greatest Common Factor (GCF) of a set of numbers is the largest number that can evenly divide each number in the set. When working with polynomials, finding the GCF involves taking into account both the coefficients and the variables. For example, in the polynomial expression \(8x^5 + 16x^4 - 20x^3 + 12\), we first focus on the coefficients, which are 8, 16, -20, and 12.
To find the GCF of these coefficients, we list the factors of each number:

• Factors of 8 are 1, 2, 4, and 8.
• Factors of 16 are 1, 2, 4, 8, and 16.
• Factors of -20 include -1, -2, -4, -5, -10, and -20, but for GCF, consider positive factors 1, 2, 4, 5, 10, 20.
• Factors of 12 are 1, 2, 3, 4, 6, and 12.

The largest number common to all these sets of factors is 4. Therefore, the GCF of the coefficients is 4.
Finding the GCF is essential because it allows us to factor the polynomial expression more efficiently, simplifying further steps.

Polynomial Expression

A polynomial is a mathematical expression consisting of variables, coefficients, and the operations of addition, subtraction, and multiplication. Each segment separated by plus or minus is called a term. For example, in the polynomial \(8x^5 + 16x^4 - 20x^3 + 12\), there are four terms. Each term consists of a coefficient (like 8, 16, -20) and a variable component (like \(x^5, x^4, x^3\)).
Polynomials play a big role in algebra and are foundational in higher-level math. Understanding polynomials helps in simplification and solving equations.
In algebra, we often rewrite polynomials in their "factorized" form to simplify or solve them. To factor a polynomial, we look for common factors in each term, like the GCF. This way, each polynomial can be expressed as the product of its GCF and another simpler polynomial.

Simplifying Expressions

Simplifying expressions involves rewriting them as simply as possible while maintaining equivalence. When factoring polynomials, one of the key steps is simplifying by factoring out the GCF.
Once we identify the GCF (in this case, 4), we divide each term of the polynomial by this GCF. This gives us another polynomial: \(2x^5 + 4x^4 - 5x^3 + 3\), which is simpler yet equivalent to the original.
The final expression is written as the product of the GCF and the simplified polynomial: \(4(2x^5 + 4x^4 - 5x^3 + 3)\). This step makes expressions easier to work with and paves the way for solving equations or further manipulation. Factoring and simplifying expressions are crucial skills in mathematics that allow for more efficient problem-solving and understanding of math concepts.