Problem 18 - Elementary Algebra

Question

In the following problems, the first quantity represents the product and the second quantity represents a factor of that product. Find the other factor. $$ 22 b^{8} c^{6} d^{3}, \quad-11 b^{8} c^{4} $$

Step-by-Step Solution

Answer

Answer: The other factor is $$-2c^{2}d^{3}$$.

Step 1: Identify the expressions

The given expression represents the product and the factor as mentioned below: Product: $$22 b^{8}c^{6}d^{3}$$ Factor: $$-11 b^{8}c^{4}$$

Step 2: Divide the product by the factor

In order to find the other factor, we need to divide the product by the given factor. So we have: $$ \frac{22 b^{8}c^{6}d^{3}}{-11 b^{8}c^{4}} $$

Step 3: Simplify the expression

Now, let's simplify the expression using exponent rules and arithmetic operations: $$ \frac{22}{-11} \times \frac{b^{8}}{b^{8}} \times \frac{c^{6}}{c^{4}} \times d^{3} $$ Starting with numbers, we have \(\frac{22}{-11}\) which becomes \(-2\). Then, observing the power of b, we have \(\frac{b^{8}}{b^{8}}\), which becomes \(b^{0}\). Any base (except 0) raised to the power of 0 equals 1. So, we have \(b^{0} = 1\). Next, we see the power of c. We have \(\frac{c^{6}}{c^{4}}\), which applying the power rule of division \((a^n / a^m = a^{n-m})\), we get \(c^{2}\). Finally, we have \(d^{3}\), as there is no corresponding term with d in the denominator.

Step 4: Combine the simplified terms

Now let's combine the simplified terms to get the other factor: $$ -2 \times 1 \times c^{2} \times d^{3} $$ So the other factor is: $$ -2c^{2}d^{3} $$

Key Concepts

Dividing Polynomials

Dividing polynomials is a key aspect of algebra. It involves breaking down complex expressions into more manageable parts. Consider the terms of the polynomials as separate entities, and remember that like terms can be divided by each other. In our example, we are dividing the polynomial expression \(22 b^{8}c^{6}d^{3}\) by \( -11 b^{8}c^{4}\). The process is similar to numerical division but requires attention to variable terms and their exponents as well.

Exponent Rules

Exponent rules are essential for simplifying expressions involving powers. When dividing terms with the same base, subtract the exponent of the denominator from the exponent of the numerator, as observed with \(\frac{c^{6}}{c^{4}} = c^{2}\). Remember, anything raised to the zero power is one, except for zero itself. Thus, \(b^{8} / b^{8} = b^{0} = 1\). These rules allow us to drastically simplify complicated expressions and find the other factor with relative ease.

Simplifying Expressions

To simplify an algebraic expression, combine like terms and reduce fractions to their simplest form. Start with any coefficients, as in our example \(\frac{22}{-11} = -2\), and then apply exponent rules to the variables. Simplification is like cleaning up a messy room; you're grouping similar items together to make the space—and the expression—more orderly and understandable.

Algebraic Factorization

Algebraic factorization involves finding the factors that, when multiplied together, give you the original expression. When you have the product and one factor, as in the given exercise, you divide the product by the known factor to find the missing piece. It's like solving a puzzle where you know the big picture and one piece, and you need to figure out what's missing. Our final factor, based on the simplified expression, is \( -2c^{2}d^{3} \). This factorization process is a vital skill and has practical applications in solving polynomial equations and inequalities.