Problem 11 - Elementary Algebra

Question

For the following problems, the first quantity represents the product and the second quantity a factor. Find the other factor. $$ 14 b^{2}+16 b, 2 b $$

Step-by-Step Solution

Answer

Question: Given the product (16x - 32) and the factor 16, find the other factor. Answer: The other factor is (x - 2).

Step 1: Identify the given product and factor

The given product is \((16x - 32)\) and the given factor is \(16\).

Step 2: Divide the product by the factor

Next, we'll divide the product by the given factor to find the other factor: $$ \frac{16x - 32}{16} $$

Step 3: Simplify the expression

Finally, simplify the expression to obtain the other factor: $$ \frac{16x}{16} - \frac{32}{16} = x - 2 $$

Step 4: Write the result

The other factor we were looking for is \((x - 2)\). So, the expression can be written as a product of these two factors: $$ 16x - 32 = {(16)(x - 2)} $$

Key Concepts

Algebraic Division

Algebraic division is the process of dividing a polynomial by another polynomial. It's akin to long division with numbers, but in this case, terms of algebraic expressions are divided to simplify the expressions. For instance, when faced with an expression such as \(16x - 32\), and you know one of the factors is \(16\), you divide the entire expression by \(16\) to find the other factor. This is done term-by-term:\[\frac{16x}{16} = x\quad\text{and}\quad\frac{32}{16} = 2\]So by dividing \(16x - 32\) by \(16\), we deduce the other factor as \(x - 2\). This method of division helps to break down complex algebraic expressions into their simplest components, making them easier to work with in further calculations.

Simplifying Expressions

Simplifying expressions means reducing them to their simplest form. It involves combining like terms, factoring, expanding expressions, or using other algebraic properties. The motive is to transform a complex expression into something more manageable. For simplification, each term of the expression should be divided by the common factor, if one exists, as we saw with the division of each term in \(16x - 32\) by \(16\). After this step, the expression \(x - 2\) is in its simplest form and there are no further common factors to divide by.Simplification is crucial as it not only makes expressions easier to understand but also lays the groundwork for solving equations efficiently. It helps in recognizing patterns and makes algebraic manipulations less error-prone.

Identifying Factors

Identifying factors in a given algebraic expression is like uncovering the building blocks of the expression. A factor is a quantity that divides another quantity exactly without leaving a remainder. In our example, we identified \(16\) as a factor of the expression \(16x - 32\), hence the expression could be written in a factored form as \(16(x - 2)\).The ability to identify factors is vital for simplifying expressions and solving equations. In factorizing algebraic expressions, one usually looks for common factors in the terms of the expression, special products, or patterns such as the difference of squares. Recognizing and extracting factors are foundational skills that enable students to tackle higher-level algebra problems, such as finding zeros of polynomials or solving quadratic equations.