Problem 60 - Elementary Algebra

Question

For the following problems, note how many: $$ \mathrm{y}^{2} \text { 's in } 3 x^{3} y^{2} ? $$

Step-by-Step Solution

Answer

Answer: 1

Step 1: Identifying the term \(y^2\) in the expression

The expression given is \(3x^3y^2\). Notice that there is a \(y^2\) term present in this expression.

Step 2: Counting the number of \(y^2\) terms

In the expression \(3x^3y^2\), there is only one term, which is a product of three factors: \(3\), \(x^3\), and \(y^2\). Hence, there is only one \(y^2\) term present in this expression. So, there is 1 occurrence of the term \(y^2\) in the expression \(3x^3y^2\).

Key Concepts

Algebraic Terms

When working with polynomial expressions, it's important to understand the concept of "algebraic terms." An algebraic term is essentially a piece of a polynomial that consists of numbers (called coefficients), variables, and sometimes exponents.
For example, in the expression \(3x^3y^2\), there are three different parts of the term:

• The number \(3\), which is the coefficient.
• The variable \(x\), raised to the power of 3, which shows the influence of \(x\) in the term.
• The variable \(y\) raised to the power of 2, represented as \(y^2\), showing the role of \(y\).

It's the combination of all these elements that form a complete algebraic term.
This term is a product of its parts and they work together to provide the expression its unique identity.

Polynomial Factoring

Polynomial factoring involves breaking down a complex polynomial expression into simpler, multiplying terms (factors) that when multiplied together give the initial polynomial. It's like breaking apart a lego build into its individual blocks.
For easier terms, sometimes the whole expression can be treated as a singular factor. However, complex polynomials often involve finding a common factor or rearranging terms.

• In our expression \(3x^3y^2\), the factors are \(3\), \(x^3\), and \(y^2\).
• Every factor contributes to the make-up of the whole expression.
• Factoring helps simplify expressions and solve equations more conveniently.

This process is essential when solving polynomial equations or simplifying expressions because it reveals possible zeros of the polynomial.

Exponents

Exponents are a crucial component in understanding polynomials, and they tell us how many times to multiply the base by itself. Essentially, an exponent is like a shorthand notation for repeated multiplication of the same number or expression.
In our example \(3x^3y^2\), we have two important exponents:

• \(x^3\) indicates that \(x\) is to be multiplied by itself 3 times: \(x \times x \times x\).
• \(y^2\) suggests that \(y\) is multiplied by itself 2 times: \(y \times y\).

Exponents greatly affect the value of algebraic expressions. They help simplify lengthy multiplications into a more manageable form.
Recognizing exponents in expressions not only simplifies that term but also provides insights into operations needed to resolve them.