Problem 6
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. $$ 16 a, 8 $$
Step-by-Step Solution
Verified Answer
Answer: The other factor is $$2a$$.
1Step 1: Identify the Product and the Given Factor
The product is $$16a$$ and the given factor is $$8$$.
2Step 2: Write the Relationship Between Product and Factors
The relationship between product and factors can be written as:
Product = Factor 1 × Factor 2
3Step 3: Substitute the Known Values
Since we have the product ($$16a$$) and one factor ($$8$$), we can substitute them in the equation:
$$16a = 8 \times \text{(Factor 2)}$$
4Step 4: Solve for the Unknown Factor
To find the unknown factor (Factor 2), we need to divide both sides of the equation by $$8$$:
Factor 2 = $$\frac{16a}{8}$$
5Step 5: Simplify the Expression
Divide $$16a$$ by $$8$$ to find the value of the unknown factor:
Factor 2 = $$2a$$
The other factor is $$2a$$.
Key Concepts
Understanding AlgebraExploring Product and FactorsSimplifying Expressions
Understanding Algebra
Algebra is a branch of mathematics where numbers and symbols come together to express general principles.
It allows us to create expressions and equations to solve problems and find unknown quantities.
When dealing with algebraic expressions, you often see variables like \(a\), which stand for unknown values or changing quantities.
The goal is to manipulate this expression to uncover the unknown factors or values.
It allows us to create expressions and equations to solve problems and find unknown quantities.
When dealing with algebraic expressions, you often see variables like \(a\), which stand for unknown values or changing quantities.
- These variables work alongside numbers, combined using various operations like addition and multiplication.
- Algebra helps simplify expressions, making complex problems easier to manage and solve.
The goal is to manipulate this expression to uncover the unknown factors or values.
Exploring Product and Factors
In mathematics, a product is the result of multiplying numbers or expressions.
A factor is a number or expression that divides the product evenly without leaving a remainder.
Understanding the relationship between product and factors is key in problems involving multiplication or division.
We have a product \(16a\) and one known factor \(8\). We are tasked to find the other factor.
By expressing the product as a multiplication of the known factor with the unknown one—\(16a = 8 \times \text{Factor 2}\)—we can rearrange this into an equation that reveals the missing part.
A factor is a number or expression that divides the product evenly without leaving a remainder.
Understanding the relationship between product and factors is key in problems involving multiplication or division.
- For the expression \(16a\), the product is formed by multiplying the two factors: Factor 1 and Factor 2.
- Given one factor, you can solve for the unknown factor by isolating it in the equation.
We have a product \(16a\) and one known factor \(8\). We are tasked to find the other factor.
By expressing the product as a multiplication of the known factor with the unknown one—\(16a = 8 \times \text{Factor 2}\)—we can rearrange this into an equation that reveals the missing part.
Simplifying Expressions
Simplifying an expression means finding an equivalent expression that is simpler or more efficient.
This involves processes like cancelling out terms, performing arithmetic operations, or factoring out common elements.
When dividing \(16a\) by \(8\), you perform a division that simplifies the expression.
Here, \(16a / 8\) simplifies to \(2a\), representing the factor we were calculating for.
By reducing \(16a\) divided by \(8\) to \(2a\), we've successfully found and simplified the missing factor of the product.
This involves processes like cancelling out terms, performing arithmetic operations, or factoring out common elements.
- It makes calculations easier, especially when working with algebraic expressions.
- In the problem, the challenge is to simplify the calculation \(\frac{16a}{8}\) to find the other factor.
When dividing \(16a\) by \(8\), you perform a division that simplifies the expression.
Here, \(16a / 8\) simplifies to \(2a\), representing the factor we were calculating for.
By reducing \(16a\) divided by \(8\) to \(2a\), we've successfully found and simplified the missing factor of the product.
Other exercises in this chapter
Problem 6
Use the grouping method to factor the following polynomials. When factoring the polynomial \(8 a^{2} b^{4}-4 b^{4}+14 a^{2}-7\) in Sample Set A, we grouped toge
View solution Problem 6
Factor \(8 m^{3}(n-4)-6 m^{2}(n-4)\).
View solution Problem 7
For the following problems, factor, if possible, the polynomials. $$ m^{4}-n^{4} $$
View solution Problem 7
Factor, if possible, the following trinomials. \(x^{2}-x y-6 y^{2}\)
View solution