Problem 11
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. $$ 8 x^{4}, 4 x $$
Step-by-Step Solution
Verified Answer
Answer: The other factor is \(2x^3\).
1Step 1: Analyze the given information
We are given the product \(8x^4\) and one of its factors: \(4x\). Our goal is to find the other factor.
2Step 2: Divide the product by the given factor
To find the other factor, we should divide the product by the given factor. In this case, we'll divide \(8x^4\) by \(4x\). To do this, divide the coefficients and subtract the exponents:
$$
\frac{8x^4}{4x} = \frac{8}{4} \cdot \frac{x^4}{x}.
$$
3Step 3: Simplify the expression
Now, we need to simplify the expression. To do this, perform the division and subtract the exponents:
$$
\frac{8}{4} \cdot \frac{x^4}{x} = 2 \cdot x^{(4-1)}.
$$
4Step 4: Write the final answer
After simplifying, our expression becomes:
$$
2 \cdot x^{(4-1)} = 2x^3.
$$
Hence, the other factor is \(2x^3\).
Key Concepts
Polynomial DivisionAlgebraic ExpressionsExponent Rules
Polynomial Division
Understanding polynomial division is like learning a new language in mathematics—it's all about breaking down complex expressions into simpler parts. It's similar to long division with numbers, but instead deals with variables and exponents. Imagine having a big chunk of clay that you need to split into equal parts; polynomial division does just that but with algebraic expressions.
When dividing polynomials, we align terms with similar degrees and divide them, just like in our exercise where we divided the polynomial term 4 with coefficient and variable term from the product. It's crucial not to get intimidated by the variables; treat them as placeholders that follow arithmetic rules. Just as dividing 8 apples between 4 people gives each person 2 apples, dividing 8x^4 by 4x leaves us with a simpler form. It’s a fundamental technique in algebra that allows students to simplify expressions and solve equations more efficiently.
When dividing polynomials, we align terms with similar degrees and divide them, just like in our exercise where we divided the polynomial term 4 with coefficient and variable term from the product. It's crucial not to get intimidated by the variables; treat them as placeholders that follow arithmetic rules. Just as dividing 8 apples between 4 people gives each person 2 apples, dividing 8x^4 by 4x leaves us with a simpler form. It’s a fundamental technique in algebra that allows students to simplify expressions and solve equations more efficiently.
Algebraic Expressions
Algebraic expressions are the alphabets to the language of algebra. They are the combinations of numbers, variables (like x or y), and operators (like addition and multiplication) that represent a quantity. Think of it as a recipe that combines various ingredients to create a dish - in this case, a mathematical result.
Algebraic expressions embody the core of algebraic thinking, transforming real-world problems into solvable equations. For instance, in our exercise, the expression 8x^4 communicates the multiplication of 8 and the variable x raised to the fourth power. When dealing with algebraic expressions, we manipulate them using operations such as addition, subtraction, multiplication, and division, always respecting the hierarchy and properties of algebra.
Algebraic expressions embody the core of algebraic thinking, transforming real-world problems into solvable equations. For instance, in our exercise, the expression 8x^4 communicates the multiplication of 8 and the variable x raised to the fourth power. When dealing with algebraic expressions, we manipulate them using operations such as addition, subtraction, multiplication, and division, always respecting the hierarchy and properties of algebra.
Exponent Rules
Exponent rules are the playbook for managing powers in algebra. They tell us how to handle multiplication, division, and raising powers to powers when it comes to expressions with exponents. For instance, when we divide exponents with the same base, we subtract their powers, just like reducing the height of a tower by taking off layers from the top.
In our example, simplification involved subtracting the exponent of the denominator from the exponent of the numerator, according to the rules for dividing exponents. Remember, these rules make calculations tidier and quicker, enabling students to crack complex problems with ease. By mastering these rules, you will be well-equipped to handle the potential complexities that come with higher-level algebra, and you’ll be able to simplify expressions like a pro.
In our example, simplification involved subtracting the exponent of the denominator from the exponent of the numerator, according to the rules for dividing exponents. Remember, these rules make calculations tidier and quicker, enabling students to crack complex problems with ease. By mastering these rules, you will be well-equipped to handle the potential complexities that come with higher-level algebra, and you’ll be able to simplify expressions like a pro.
Other exercises in this chapter
Problem 11
For the following problems, use the grouping method to factor the polynomials. Some polynomials may not. be factorable using the grouping method. $$ a r+4 a s+5
View solution Problem 11
For the following problems, factor the polynomials. $$ 4 x-6 $$
View solution Problem 12
For the following problems, factor the trinomials when possible. $$ x^{2}+6 x+5 $$
View solution Problem 12
Factor \(6 x^{2}+7 x-3\)
View solution