Problem 45 - Intermediate Algebra

Question

In the following exercises, translate the phrases into algebraic expressions. (a) the sum of \(4 a b^{2}\) and \(3 a^{2} b\) (b) the product of \(4 y^{2}\) and \(5 x\) (c) Fifteen more than \(m\) (d) \(9 x\) less than \(121 x^{2}\)

Step-by-Step Solution

Answer

(a) \(4ab^2 + 3a^2b\), (b) \(20xy^2\), (c) \(m + 15\), (d) \(121x^2 - 9x\)

Step 1: Translate phrase (a)

Identify the key components in the phrase 'the sum of \(4ab^2\) and \(3a^2b\)'. The word 'sum' indicates addition. So, you write the algebraic expression as: \[ 4ab^2 + 3a^2b \]

Step 2: Translate phrase (b)

Identify the key components in the phrase 'the product of \(4y^2\) and \(5x\)'. The word 'product' indicates multiplication. So, you write the algebraic expression as: \[ 4y^2 \times 5x \] Then simplify: \[ 20xy^2 \]

Step 3: Translate phrase (c)

Identify the key components in the phrase 'Fifteen more than \(m\)'. The phrase 'more than' indicates addition. So, you write the algebraic expression as: \[ m + 15 \]

Step 4: Translate phrase (d)

Identify the key components in the phrase '\(9x\) less than \(121x^2\)'. The phrase 'less than' indicates subtraction. So, you write the algebraic expression as: \[ 121x^2 - 9x \]

Key Concepts

Sum in Algebra

In algebra, the term 'sum' refers to the result of addition. It's a fundamental concept where we combine two or more quantities. When we talk about the sum of algebraic expressions, we are essentially adding these expressions together. For example, consider the expression in part (a) of the original exercise: the sum of \(4ab^{2}\) and \(3a^{2}b\). Here, 'sum' tells us to add the two parts: \( 4ab^{2} + 3a^{2}b \). By combining these terms, we get a single expression that represents their sum.

Product in Algebra

In algebra, the term 'product' refers to the result of multiplication. Understanding how to find the product is crucial for various algebraic problems. In part (b) of the exercise, we are asked to find the product of \(4y^{2}\) and \(5x\). 'Product' indicates multiplication, so we multiply these two expressions: \(4y^{2} \times 5x\). Simplifying the multiplication, we get: \(20xy^{2}\). This is the product of the expressions, showing how the two terms are multiplied to get a new algebraic expression.

Addition in Algebra

Addition in algebra involves combining two or more expressions to get their total. This concept is very similar to basic arithmetic addition, but now it includes variables. Take part (c) of the exercise as an example: 'Fifteen more than \(m\)'. The phrase 'more than' indicates addition. This means we add 15 to the variable \(m\), resulting in the expression: \(m + 15\). This makes it clear how we can translate verbal phrases into algebraic expressions using addition.

Subtraction in Algebra

Subtraction in algebra is about taking one quantity away from another. It's an essential operation in simplifying and solving equations. In part (d) of the exercise, the problem states: '\(9x\) less than \(121x^{2}\)'. The phrase 'less than' means we need to subtract \(9x\) from \(121x^{2}\). Therefore, the algebraic expression becomes \(121x^{2} - 9x\). Through this, we see how subtraction helps us simplify and handle different algebraic expressions effectively.