Problem 32 - Elementary Algebra

Question

Factor the greatest common factor from each polynomial. \(10 q^{2}+14 q+20\)

Step-by-Step Solution

Answer

2(5q^2 + 7q + 10)

Step 1: Identify the coefficients

List the coefficients of each term in the polynomial. For the polynomial given, the coefficients are 10, 14, and 20.

Step 2: Find the greatest common factor (GCF) of the coefficients

Determine the GCF of the coefficients (10, 14, and 20). The GCF of 10, 14, and 20 is 2.

Step 3: Factor out the GCF

Divide each term in the polynomial by the GCF (2) and factor it out. The polynomial becomes: \[ 10q^2 + 14q + 20 = 2(5q^2 + 7q + 10) \]

Step 4: Verify the factored form

Multiply the GCF back through the factored polynomial to ensure it matches the original polynomial.\[ 2(5q^2 + 7q + 10) = 10q^2 + 14q + 20 \]This confirms that the factored form is correct.

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is a key concept in algebra. It helps simplify polynomials by factoring out common values. To find the GCF, look for the largest number that can divide all the coefficients without leaving a remainder. For example, consider the coefficients 10, 14, and 20. The factors of 10 are 1, 2, 5, and 10. The factors of 14 are 1, 2, 7, and 14. The factors of 20 are 1, 2, 4, 5, 10, and 20.

The common factor among them is 2. So, we use 2 to simplify the polynomial.

Coefficients

Coefficients are the numerical parts of the terms in a polynomial. In the polynomial 10q² + 14q + 20, the coefficients are 10, 14, and 20. Identifying the coefficients helps in determining the GCF and simplifying the polynomial.

Here’s how:

• Identify all numerical coefficients in the polynomial.
• Use these coefficients to find the GCF.

This understanding is essential for factoring polynomials and is the first step in many algebraic operations.

Polynomial Factoring

Polynomial factoring involves expressing a polynomial as a product of its factors. After finding the GCF, divide each term by the GCF and factor it out. For the polynomial 10q² + 14q + 20, we factor out the GCF (2):

\[ 10q^2 + 14q + 20 = 2(5q^2 + 7q + 10) \]

This step turns a complex polynomial into a more manageable form, making further calculations or integrations easier. It is an essential skill in algebra.

Algebra

Algebra encompasses a wide range of mathematical concepts, including polynomial factoring. Mastering these concepts involves understanding and manipulating numbers and variables. Polynomial factoring, for instance, requires knowledge of:

• Coefficients
• Greatest Common Factor
• Basic arithmetic operations

In algebra, the ability to simplify expressions, like turning 10q² + 14q + 20 into 2(5q² + 7q + 10), is crucial. It lays the foundation for solving equations, calculus, and more advanced math topics. Practice these skills to get comfortable with algebra.