Problem 16 - Elementary Algebra

Question

In the term \(10 a b^{4}\), the coefficient of (a) \(a b^{4}\) is _________. (b) \(b^{4}\) is ________. (c) \(a\) is _________. (d) 10 is _________. (e) \(10 a b^{3}\) is _________.

Step-by-Step Solution

Answer

Question: Find the coefficients of the following in the term \(10ab^4\): a) Coefficient of \(ab^4\) b) Coefficient of \(b^4\) c) Coefficient of \(a\) d) Coefficient of 10 e) Coefficient of \(10ab^3\) Answer: a) 10 b) \(10a\) c) \(10b^4\) d) 1 e) \(b\)

Step 1: (a) Coefficient of \(ab^4\)

The term is already \(10ab^4\). This means that the coefficient of the whole term \(ab^4\) is 10.

Step 2: (b) Coefficient of \(b^4\)

We are looking for the coefficient of \(b^4\) in the term \(10ab^4\). We need to isolate \(b^4\) by dividing the term by \(a\). So, the coefficient of \(b^4\) is \(10a\).

Step 3: (c) Coefficient of \(a\)

We are looking for the coefficient of \(a\) in the term \(10ab^4\). We need to isolate \(a\) by dividing the term by \(b^4\). So, the coefficient of \(a\) is \(10b^4\).

Step 4: (d) Coefficient of 10

The term is \(10ab^4\). Since 10 is a constant, it doesn't have a coefficient itself. We can consider its coefficient as 1 for the purpose of this exercise.

Step 5: (e) Coefficient of \(10ab^3\)

We are given the term \(10ab^4\) and need to find the coefficient of \(10ab^3\). To do this, we need to divide the term by \(b\). So, the coefficient of \(10ab^3\) is \(b\).

Key Concepts

Algebraic Expressions

An algebraic expression is a collection of numbers, variables, and mathematical operations like addition, subtraction, multiplication, and division. It's essentially a phrase that represents a particular quantity or relationship. For example, the expression 2x + 3 indicate that we have 2 times some unknown number x added to 3.

In algebraic expressions, we frequently encounter coefficients. A coefficient is a number that multiplies a variable. For instance, in the term 10ab^4, 10 is the coefficient for the whole term, telling us how many times we have the product of a and b^4. Coefficients can be positive or negative and sometimes can be fractions or decimals, depending on the expression. Understanding coefficients is crucial to manipulating and solving algebraic expressions.

Polynomial Terms

Polynomial terms are the building blocks of polynomials, which are algebraic expressions consisting of variables raised to non-negative integer powers and their coefficients. In a polynomial term such as 10ab^4, three components are visible: the coefficient (10), the variable a, and the variable b raised to the fourth power. This term can be part of a larger polynomial, such as 10ab^4 - 5a + 3.

Each term in a polynomial is separate and distinct, often combined by addition or subtraction. When trying to find the coefficient of a particular part of the term, we can manipulate the term to isolate that part. For example, to find the coefficient of b^4 in 10ab^4, we treat the variable a as part of the coefficient, resulting in 10a as the coefficient of b^4.

Elementary Algebra

Elementary algebra is the branch of mathematics that deals with the properties of operations and the constructions derived from them, including polynomials, equations, and algebraic structures. Core to this subject is the understanding of variables, coefficients, and the rules that govern mathematical operations on them.

In our textbook problem, applying the principles of elementary algebra allows us to decipher the coefficient of different parts of the term 10ab^4. This involves recognizing which part of the term we're focusing on and performing the necessary operation, like division, to isolate the coefficient. By studying elementary algebra, students develop the ability to work with unknowns and solve for variables, which is a fundamental skill in higher mathematics and many real-world problems.