91Ó°ÊÓ

When multiplying polynomials, one of the crucial steps involves the coefficient multiplication. A coefficient is a number that is multiplied by a variable or variables in an algebraic expression. For example, in the expression \( -5t^3 \), the coefficient is \( -5 \). When multiplying two terms, each with its own coefficient, as seen in the exercise with \( -5t^3 \) and \( 2t^4 \), the first step is to multiply these coefficients directly: \( -5 \times 2 = -10 \).

This step is fundamental since it sets the groundwork for the overall multiplication process. The coefficient resulting from this multiplication impacts the entire term's value, and any mistake here will carry through subsequent steps.

Once the coefficients are multiplied, the next critical operation is exponent addition. In algebra, when we multiply terms with the same base, we add the exponents. This rule is based on the properties of exponents, which state that \( t^m \times t^n = t^{m+n} \). In our exercise, we have \( t^3 \times t^4 \), which simplifies to \( t^{3+4} \), or \( t^7 \).

This step highlights the importance of understanding exponent rules to successfully manipulate and simplify algebraic expressions. When adding exponents, it’s important to ensure that the terms have the same base and that only the exponents are summed, leaving the base unchanged.

The term algebraic expressions refers to a combination of numbers, variables, and arithmetic operations. In the context of our exercise, \( -5t^3 \) and \( 2t^4 \) are both algebraic expressions. Multiplying them, as we've learned through coefficient multiplication and exponent addition, leads to a new algebraic expression: \( -10t^7 \).

It is essential to understand the structure of algebraic expressions and the rules that govern them. Mastery of these rules helps in the accurate and simplified presentation of the expressions, which is a cornerstone of algebra. For example, recognizing that the base 't' is the same in both terms and thus the exponents can be added is fundamental in simplifying the expression to its final form.