Problem 7 - Elementary Algebra

Question

For the following problems, factor the polynomials. $$ 9 a+18 $$

Step-by-Step Solution

Answer

Answer: The factored form of the polynomial \(9a + 18\) is \(9(a + 2)\).

Step 1: Identify the coefficients of the polynomial terms

In the given polynomial, \(9a + 18\), the coefficients are 9 (for the term \(9a\)) and 18 (for the term \(18\)).

Step 2: Find the greatest common divisor (GCD) of the coefficients

The coefficients of the terms are 9 and 18. To find their GCD, consider their prime factors: - The prime factors of 9 are \(3 \times 3\). - The prime factors of 18 are \(2 \times 3 \times 3\). The common prime factors are \(3 \times 3\) or \(3^2\). Therefore, the GCD of 9 and 18 is \(3^2 = 9\).

Step 3: Divide the polynomial by the GCD

We found the GCD of the coefficients to be 9. Now, divide both terms of the polynomial by 9: - \(9a \div 9 = a\) - \(18 \div 9 = 2\) So, the new polynomial becomes \(a + 2\).

Step 4: Write the polynomial in its factored form

To write the polynomial in the factored form, we will put the GCD and the new polynomial as factors. Factored polynomial: \(9(a + 2)\) The factored form of the given polynomial \(9a + 18\) is \(9(a + 2)\).

Key Concepts

Greatest Common Divisor (GCD)

The greatest common divisor (GCD) is the largest number that divides two or more numbers without leaving a remainder. Finding the GCD is crucial when factorizing polynomials, as it helps to simplify expressions.

To determine the GCD, you should:

• List the prime factors of each number.
• Identify the common prime factors.
• Multiply these common factors together to get the GCD.

For example, in the polynomial problem where we have the terms 9 and 18, the prime factors of 9 are \(3 \times 3\), and those of 18 are \(2 \times 3 \times 3\). The common prime factors are \(3 \times 3\), which means the GCD is 9.

Using the GCD to factor polynomials makes expressions simpler and easier to work with.

Prime Factorization

Prime factorization breaks down a number into its smallest building blocks, which are prime numbers. A prime number is only divisible by 1 and itself. This process is crucial in finding the greatest common divisor.

Let's say you want to factorize 18:

• Divide 18 by the smallest prime number, 2, to get 9.
• Since 9 isn't divisible by 2, try dividing it by the next smallest prime, which is 3.
• Divide by 3 to get 3, and then divide by 3 again to get 1.

Therefore, the prime factorization of 18 is \(2 \times 3 \times 3\).

Prime factorization is used to simplify fractions, find GCD, and break down algebraic expressions into manageable parts when solving polynomial problems.

Algebraic Expressions

Algebraic expressions combine numbers, variables, and operations. Understanding how to manipulate these expressions is fundamental in algebra.

When you have an expression like \(9a + 18\), it is made up of terms, coefficients, and variables.

• Terms: The individual parts of an expression separated by a plus (+) or minus (−) sign, like \(9a\) and 18.
• Coefficients: The numerical part of a term that is multiplied by the variable, such as the 9 in \(9a\).
• Variables: Symbols that represent unknown values, like \(a\).

Factoring these expressions involves finding common factors and rewriting the expression in its simplest form, making it easier to understand and solve problems.

Processed correctly, algebraic expressions become the building blocks for more complex mathematics, leading to solutions of equations and understanding of functions.