Problem 19 - Elementary Algebra

Question

For the following problems, the first quantity represents the product and the second quantity a factor. Find the other factor. $$ 26 b^{2}+13 b, 13 b $$

Step-by-Step Solution

Answer

Answer: The unknown factor is (2b + 1).

Step 1: Rewrite the Equation

Our goal is to rewrite the given expression in the form of a product involving the known factor 13b: $$ (26b^2 + 13b) = (13b) \times (unknown\ factor) $$

Step 2: Factor the Known Factor 13b out of the Expression

We need to factor 13b out of the given expression to find the unknown factor: $$ (26b^2 + 13b) = (13b)(2b+1) $$ Notice that each term in the expression is divisible by 13b (26b^2 ÷ 13b = 2b and 13b ÷ 13b = 1).

Step 3: Identify the Other Factor

Now we have the expression in the desired form, and we can identify the other factor: $$ (26b^2 + 13b) = (13b)(2b+1) $$ So, the unknown factor is \((2b+1)\).

Key Concepts

Polynomial Expressions

Polynomial expressions are mathematical expressions that consist of variables, coefficients, and exponents combined using addition, subtraction, and multiplication. These expressions can have one or more terms, where each term is made up of a constant multiplied by a variable raised to a power.

• A single term example: \( 3x^2 \)
• A multi-term example: \( 4x^3 - 2x^2 + 7 \)

Understanding polynomial expressions is essential because they form the basis of many algebraic equations and operations. They are often simplified or manipulated to solve real-world problems. The complexity of a polynomial is determined by its degree, which is the highest exponent of the variable in the expression.The given expression in the original exercise, \(-x - 2y - c^2\), consists of three terms, each with different variables and no shared coefficients, which makes it a polynomial expression too.

Factoring Techniques

Factoring is the process of breaking down a complex expression into simpler factors that, when multiplied, give the original expression. This technique is crucial in algebra as it simplifies many problems, making them easier to solve.To factor a polynomial, follow these general steps:1. **Identify a Common Factor**: Look for any common factor present in all terms of the polynomial. In our earlier example \((26b^2 + 13b)\), the common factor is \(13b\).2. **Divide Each Term**: Rewrite each term by dividing it by the common factor. In the example, \(26b^2\) becomes \(2b\) and \(13b\) becomes \(1\).3. **Write the Expression as a Product**: After factoring out the common factor, the expression should be written in a simple multiplication format. In our case, the factored form is \((13b)(2b+1)\).Using these steps, the expression simplifies revealing the unknown factor, significantly easing further calculations or solving.

Algebraic Multiplication

Algebraic multiplication, an important skill in mathematics, involves multiplying different algebraic expressions. It relies on understanding how to distribute and combine terms.When multiplying expressions, follow these critical rules:

• **Distributive Property**: Multiply each term in the first expression by each term in the second expression. For example, \((a+b)(c+d) = ac + ad + bc + bd\).
• **Combine Like Terms**: After distribution, ensure that similar terms (terms with the same variables and exponents) are combined.

In algebraic multiplication, accuracy and attention to detail during distribution and combination ensure the correct result. The goal is to simplify the expression to make it manageable and understandable.Returning to our factorized expression, \((13b)(2b+1)\), once you multiply the factors back using these principles, you receive the original expression, confirming the correctness of the factorization process. This underscores the importance of mastering algebraic multiplication for effective problem-solving in algebra.