Problem 9 - Elementary Algebra

Question

For the following problems, the first quantity represents the product and the second quantity a factor. Find the other factor. $$ 16 a+64,8 $$

Step-by-Step Solution

Answer

Answer: The other factor is \(2(a+4)\).

Step 1: Simplify the expression

First, we can factor out the common constant factor, which is \(16\), from the expression: $$ 16a + 64 = 16(a + 4) $$ Now, we have the expression in the form of \(16(a+4)\).

Step 2: Divide by the given factor

We are given that one of the factors is \(8\), therefore we need to divide the expression \(16(a+4)\) by \(8\) to find the other factor: $$ \frac{16(a+4)}{8} $$

Step 3: Simplify the division

Now, we will simplify the division by cancelling out the common factors: $$ \frac{16(a+4)}{8} = 2(a+4) $$

Step 4: The other factor

The other factor is the simplified result of the division \(2(a+4)\). So, the other factor is: $$ 2(a+4) $$

Key Concepts

Polynomial Division

Polynomial division is similar to the long division we learned in arithmetic but applied to polynomials instead of numbers. Imagine cutting a pizza and each piece is a part of a polynomial. We divide one polynomial by another to see how many times it fits completely or what remains. In our exercise, think of the polynomial you are dividing by as the "divisor" and the original polynomial as the "dividend." The goal is to find out the missing "other factor."

• Start by dividing the leading term of the dividend by the leading term of the divisor. This gives the first term of your answer.
• Then multiply this term by the entire divisor and subtract it from the dividend.
• Continue the process with the result until you cannot divide anymore.

Notice how we break things down? As with our example, the division of expressions has been simplified step-by-step:

• Write the original multiplied expression or product.
• Recognize the known factor (divisor).
• Divide to find the other missing factor.

Simplification

Simplification is all about making expressions easier to work with without changing their value. When you see complex algebraic expressions, your main task is to make them simpler. This means finding and eliminating any unnecessary parts or reducing fractions where possible.Firstly, we identify similar terms or coefficients in an expression. Simplifying involves steps like:

• Cancelling common factors: Divide both numerator and denominator by the same common factors.
• Combining like terms: Add or subtract terms with the same variable and exponent.

In our exercise, simplification occurred when we divided by a given factor. The expression was 16 multiplied by \((a + 4)\), which then divided down by 8, became simpler:

• The division of 16 by 8 yields 2.
• The expression is now in its reduced form: \(2(a+4)\).

Thanks to simplification, we change a complex multiplicative relationship into a clearer and more manageable expression.

Algebraic Expressions

Algebraic expressions are like sentences in math that involve numbers, variables, and operations. They don't have an equals sign like equations, making them flexible. In our example, expressions can include:

• Variables - symbols like \(x\) or \(y\) that stand in for numbers.
• Constants - regular numbers that don’t change, like 12 or -3.
• Operations - things like addition, subtraction, multiplication, and division.

To solve problems involving expressions, you often need to manipulate these components. With careful steps to identify products and factors, a polynomial like \(12x^{n+6}y^{2n-5}\) can be understood more easily by working with its parts. Its components (the terms and factors) reveal themselves when you isolate and simplify, allowing you to tackle more complex problems. Remember, algebraic expressions bridge the gap between arithmetic and algebra and are foundational for learning further advanced mathematics.