Problem 14 - 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. $$ 9 a^{4} b^{5}, 9 a^{4} $$

Step-by-Step Solution

Answer

Answer: $$b^{5}$$.

Step 1: Identify the given product and factor

We are given the following product and one factor: Product: $$9a^{4}b^{5}$$ Factor: $$9a^{4}$$

Step 2: Divide the product by the given factor

To find the missing factor, we will divide the product by the given factor: $$ \frac{9a^{4}b^{5}}{9a^{4}} $$

Step 3: Simplify the fraction by canceling common terms

Both the numerator and the denominator have common factors which we can cancel out. In this case, we can cancel out $$9$$ and $$a^{4}$$: $$ \frac{9a^{4}b^{5}}{9a^{4}} = \frac{9}{9}\cdot\frac{a^{4}}{a^{4}}\cdot b^{5} = 1\cdot 1\cdot b^{5} $$

Step 4: Write the result

After simplifying the fraction, we are left with the other factor: $$ b^{5} $$ So the missing factor is $$b^{5}$$.

Key Concepts

Dividing Polynomials

Dividing polynomials is similar to long division in arithmetic. It involves breaking down a complex polynomial into simpler parts by dividing it by another polynomial, known as the divisor. To do this, we align the terms in descending order of their degrees and divide the highest degree term of the dividend by the highest degree term of the divisor. We then multiply the entire divisor by the result and subtract it from the dividend, repeating the process with the remainder until we cannot divide further. It’s essential to understand that like numbers, polynomials can also be factored or broken down into more manageable pieces.

For students, it's important to remember to keep terms organized to avoid confusion. Factors like missing terms, or the need to add zero placeholders for missing degrees, should be accounted for. Mastering polynomial division helps to simplify algebraic expressions and solve complicated algebra equations.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves reducing them to their simplest form while retaining their original value. This process typically involves combining like terms, which are terms that have the same variables raised to the same power, and carrying out arithmetic operations while adhering to the order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

When simplifying, it's helpful to:

• Combine like terms by adding or subtracting coefficients
• Use the distributive property to eliminate parentheses
• Apply the laws of exponents to simplify powers and roots correctly

By doing so, the expression becomes more manageable and sets the stage for further operations, such as solving equations or evaluating the expression for given variable values.

Exponents and Powers

Exponents and powers are a way to represent repeated multiplication of the same factor. In an expression like \(a^n\), \(a\) is the base and \(n\) is the exponent, indicating that \(a\) is multiplied by itself \(n\) times. There are important rules or laws for working with exponents:

• The product of powers rule states that when multiplying two powers with the same base, add the exponents: \(a^m \times a^n = a^{m+n}\).
• The quotient of powers rule says that when dividing two powers with the same base, subtract the exponents: \(a^m \div a^n = a^{m-n}\), as long as \(a\) is not zero.
• For any non-zero base, any term raised to the power of zero is 1: \(a^0 = 1\).

Understanding these rules will make simplifying expressions with exponents much easier and help avoid common mistakes.

Algebraic Factors

Algebraic factors are expressions that can be multiplied to form a polynomial or another algebraic expression. Factoring is the process of breaking down an expression into a product of simpler expressions, which reveals the expression's building blocks. For example, \(6x^2 + 9x\) can be factored into \(3x(2x + 3)\), illustrating that 3x is a common factor.

Factors are incredibly useful for solving polynomial equations, simplifying expressions, and understanding the properties of the expression, such as its zeros or roots. When we divide a polynomial by one of its factors, we find another factor, completing the factorization process. One tip when factoring is always to look for the greatest common factor first, to simplify the expression as much as possible before using other methods like grouping or special factorization formulas.