Problem 44 - Elementary Algebra

Question

For the following problems, the first quantity represents the product and the second quantity a factor. Find the other factor. $$ 64 a^{5} b^{3} c^{11}+56 a^{4} b^{4} c^{10}-48 a^{3} b^{5} c^{9}-8 a^{3} b^{2} c^{5}, \quad 8 a^{3} b^{2} c^{5} $$

Step-by-Step Solution

Answer

Answer: The other factor is $8a^{2} b c^{6} + 7a b^{2} c^{5} - 6b^{3} c^{4} - 1$.

Step 1: Rewrite the problem as a division.

Let's rewrite the problem as a division of two expressions: $$ \frac{64a^{5} b^{3} c^{11} + 56a^{4} b^{4} c^{10} - 48a^{3} b^{5} c^{9} - 8a^{3} b^{2} c^{5}}{8a^{3} b^{2} c^{5}} $$

Step 2: Simplify the expression.

We can simplify the expression by dividing each term of the numerator by the denominator, which is the second quantity: $$ \frac{64a^{5} b^{3} c^{11}}{8a^{3} b^{2} c^{5}} + \frac{56a^{4} b^{4} c^{10}}{8a^{3} b^{2} c^{5}} - \frac{48a^{3} b^{5} c^{9}}{8a^{3} b^{2} c^{5}} - \frac{8a^{3} b^{2} c^{5}}{8a^{3} b^{2} c^{5}} $$

Step 3: Divide each term.

Divide each term in the expression: $$ 8a^{2} b c^{6} + 7a b^{2} c^{5} - 6b^{3} c^{4} - 1 $$ So, the other factor is: $$ 8a^{2} b c^{6} + 7a b^{2} c^{5} - 6b^{3} c^{4} - 1 $$

Key Concepts

Polynomial Division

Polynomial division is a technique used in algebra to divide one polynomial, which is the dividend, by another polynomial, the divisor. It works in a similar fashion to long division with numbers. When approaching an exercise involving polynomial division, we start by arranging both the dividend and the divisor in descending power order.

For instance, in our provided exercise, we divide the polynomial expression \(64a^{5}b^{3}c^{11} + 56a^{4}b^{4}c^{10} - 48a^{3}b^{5}c^{9} - 8a^{3}b^{2}c^{5}\) by the monomial \(8a^{3}b^{2}c^{5}\). The first step is to divide the leading term of the dividend \(64a^{5}b^{3}c^{11}\) by the leading term of the divisor \(8a^{3}b^{2}c^{5}\). This gives us the first term of the quotient.

Each following term of the dividend is then divided by the divisor in the same manner. This step-by-step process continues until all terms have been divided. The result of these individual divisions is the complete quotient, which is the other factor that we seek.

Remember, when dividing variables, you subtract the exponents of like bases, provided the base and the exponent of the divisor are not greater than those of the corresponding term in the dividend.

Simplifying Expressions

Simplifying expressions is a foundational skill in algebra that involves reducing an expression to its simplest form. This process makes the expression easier to understand and work with. When simplifying, we combine like terms, factor expressions, and cancel common terms.

In relation to our exercise, after performing polynomial division, we simplify the expression. This is done by dividing each term of the polynomial by the monomial factor. Here are the steps we take:

• Identify and divide the coefficients of the terms.
• Subtract the powers of like variables, following the rule that when you divide powers with the same base, you subtract the exponents.
• Write down the simplified term obtained from each division.

The simplified expression is the result of combining all the simplified terms. An essential point to note is that simplification might involve factoring expressions to cancel out common terms, which is another aspect of simplifying expressions.

Algebraic Factors

Algebraic factors are the building blocks of polynomial expressions, just like prime numbers are the building blocks of natural numbers. A factor of a polynomial is any expression that divides the polynomial without leaving a remainder. In algebraic terms, if a polynomial \(P(x)\) can be written as \(Q(x) \times R(x)\), where \(Q(x)\) and \(R(x)\) are also polynomials, then \(Q(x)\) and \(R(x)\) are factors of \(P(x)\).

In our exercise, the second quantity \(8a^{3}b^{2}c^{5}\) is a factor of the first quantity. This means the first polynomial can be evenly divided by the second, leaving us with another polynomial, which is the other factor. Understanding this concept is crucial because it simplifies complex algebraic expressions into products of simpler ones, which can be more easily solved or further factored. Identifying algebraic factors is not only a key skill for simplifying expressions but also serves as a foundation for more advanced topics like finding zeros of polynomials or solving algebraic equations.