91Ó°ÊÓ

When dealing with algebra, it's essential to understand what an algebraic expression is. An algebraic expression is a mathematical phrase that can include numbers, variables (such as x or y), and operation symbols like addition (+), subtraction (-), multiplication (×), and division (÷). For example, the expression 7x³ + 5 is an algebraic expression that indicates 7 times the cube of a variable x added to 5.

Let's revisit our exercise (7x³+5)(x²-2). Here, we have two algebraic expressions being multiplied. Each expression is made up of terms—the first has the terms 7x³ and 5, while the second has x² and -2. Polynomial multiplication is the process of multiplying these algebraic expressions together, term-by-term, using various algebra rules and properties.

To get a solid grasp on multiplication of algebraic expressions, it's also important to be comfortable with variables raised to different powers, understanding that variables with the same base can be manipulated using exponent rules, which we will discuss further in another section.

The distributive property is a cornerstone of algebra and is used to simplify and solve equations. It states that multiplying a sum or difference by a number is the same as multiplying each addend by the number and then adding or subtracting the products. In symbolic terms, if a, b, and c are any real numbers, then a(b + c) = ab + ac and a(b - c) = ab - ac.

In the context of our exercise, the distributive property allows us to multiply the terms in one polynomial by each term in the second polynomial. For the given exercise, we first take the term 7x³ of the first polynomial and distribute it over the two terms of the second polynomial, leading to 7x³×x² - 7x³×2. In the next step, we take the term 5 and repeat the distribution process, thus giving us 5×x² - 5×2.

Using the distributive property breaks down complex multiplication into simpler components, making it easier to handle and calculate the result of polynomial multiplication step by step.

What are Exponent Rules?

Exponent rules, also known as the laws of exponents, are guidelines that describe how to handle exponents in mathematical expressions, particularly when multiplying powers with the same base or raising power to a power. For instance, when we multiply two exponential expressions with the same base, we add the exponents: x^m×xⁿ = x^m+n.

In our exercise, when we distributed 7x³ with x², we applied the exponent rule mentioned above to combine the like bases. This provided us with 7x⁵, because 3 + 2 = 5. This operation is crucial for working with polynomials and simplifying algebraic expressions.

Understanding and correctly applying these rules is indispensable when multiplying polynomials or simplifying expressions with variables raised to powers. Therefore, a firm grasp of exponent rules will not only assist you in polynomial multiplication but also in a wide array of algebraic problems.