Problem 38 - Elementary Algebra

Question

For the following problems, the first quantity represents the product and the second quantity a factor. Find the other factor. $$ 10 x^{3}-35 x^{2}, \quad 5 x^{2} $$

Step-by-Step Solution

Answer

To find the other factor when given the product, \(10x^3 - 35x^2\), and one factor, \(5x^2\), we performed polynomial division and found the other factor to be \(2x\).

Step 1: Set up the division

The expression for the product is \(10x^3 - 35x^2\) and one of the factors is \(5x^2\). To find the other factor, we will divide the product by the given factor. So, we will set up the polynomial division as follows: $$\frac{10x^3 - 35x^2}{5x^2}$$

Step 2: Perform polynomial division

By dividing \(10x^3\) by \(5x^2\), we find the first term in the quotient: $$\frac{10x^3}{5x^2} = 2x$$ Now multiply \(2x\) by \(5x^2\) and subtract the result from \(10x^3 - 35x^2\): $$(10x^3 -35x^2) - (2x \cdot 5x^2) = 10x^3 - 35x^2 - 10x^3 = -35x^2$$ Since there aren't any other terms in the dividend, we stop here.

Step 3: Write down the other factor

The quotient we found in step 2, \(2x\), is the other factor we were looking for. Remember, we were given that the product is \(10x^3 - 35x^2\) and one of the factors is \(5x^2\). When we divided the product by \(5x^2\), we found the other factor to be \(2x\). So, the other factor is: $$2x$$

Key Concepts

Factorization

Factorization is a crucial concept in algebra. It involves breaking down an expression into simpler parts, called factors, that when multiplied together give back the original expression. Let's consider the product, which is a combination of multiplication operations, of the given polynomial in the exercise:

• We have a product: \(10x^3 - 35x^2\).
• One of the factors is given as \(5x^2\).
• Our goal is to determine the other factor.

The factorization process allows us to simplify polynomials and solve algebraic equations more easily. By expressing these polynomials as products of simpler expressions, we not only work towards finding one missing factor but can also simplify computational work. You'll often use factorization to solve quadratic equations, find the greatest common factor, and even work within calculus applications. To factor the expression given, we divide the entire product by the known factor to find the missing piece.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation signs. In the context of our problem, we are working with a polynomial, which is a specific type of algebraic expression composed of variables raised to whole number exponents. Here is what makes up an algebraic expression:

• Terms: Each part of an expression separated by plus or minus signs. In \(10x^3 - 35x^2\), there are two terms: \(10x^3\) and \(-35x^2\).
• Coefficients: These are the numerical parts of terms. In \(10x^3\), 10 is the coefficient.
• Variables: These are symbols that represent numbers. In this case, \(x\) is the variable.
• Exponents: They indicate how many times the variable is multiplied by itself. In \(x^3\), 3 is the exponent.

Working with algebraic expressions involves operations such as addition, subtraction, multiplication, and division. By manipulating these expressions, you can solve equations, simplify complex expressions, and delve deeper into advanced topics such as polynomial division, which is specially used in this exercise.

Quotient Determination

Quotient determination is the main objective when dividing polynomials. It involves finding what remains, or the quotient, after one polynomial is divided by another. In our exercise:

• We start with the dividend \(10x^3 - 35x^2\).
• We divide this by the divisor \(5x^2\).

The process of dividing one polynomial by another is similar to numerical division but involves carefully handling the variables and their exponents. Here's how we approached the problem:
1. **Divide the First Term:** Take the first term of the dividend \(10x^3\) and divide it by the first term of the divisor \(5x^2\). This yields \(2x\). 2. **Multiply and Subtract:** Multiply the entire divisor \(5x^2\) by the quotient term \(2x\) to get \(10x^3\). Subtract this result from the original dividend to eliminate the first term, leaving us with \(-35x^2\). 3. **Repeat:** In this problem, there are no remaining terms, so \(2x\) is the complete quotient, and thus our other factor.Finding the quotient is key to many algebraic processes, allowing us to solve polynomial equations and factor them into simpler components. With practice, quotient determination through polynomial division becomes a valuable skill in algebra.