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To understand the multiplication of polynomials, it is crucial to first grasp the concept of algebraic expressions. In algebra, an expression is a combination of numbers, variables, and operators. Variables are symbols used to represent unknown values and are often denoted by letters such as \( x \), \( y \), and \( z \). An algebraic expression can be as simple as a single number or variable, or it can be more complex, involving multiple terms and operations like addition and multiplication. Terms in an algebraic expression are usually separated by addition or subtraction signs.
A polynomial is a specific type of algebraic expression that includes constants, variables, and exponents. For example, \( 3x^2 + 4x - 7 \) is a polynomial with three terms, where each term consists of a coefficient (like 3, 4, and -7 in this example) and a variable raised to a power (like \( x^2 \) and \( x \)). Understanding this structure is essential for performing operations like multiplication on polynomials.

The distributive property is a fundamental principle in algebra that allows us to simplify expressions. It states that for any numbers \( a \), \( b \), and \( c \), the expression \( a(b + c) \) equals \( ab + ac \). This property can be used to multiply a single term by each term in a sum separately. When it comes to polynomials, the distributive property lets us distribute one polynomial over another, term by term.
In the given exercise, \( (3x^2 + 4x - 7)(2x^2) \), we distribute \( 2x^2 \) to each term in the polynomial \( 3x^2 + 4x - 7 \). This is done by multiplying \( 2x^2 \) with each term separately, following the property \( a(b + c + d) = ab + ac + ad \). This approach simplifies complex polynomial multiplications and ensures that no terms are missed.

Multiplying polynomials involves using both the structure of algebraic expressions and the distributive property. In practice, it means distributing each term of one polynomial to every term of the other polynomial. It requires attention to both coefficients and exponents.
Let's break this down using the exercise provided: the expression \( (3x^2 + 4x - 7)(2x^2) \) illustrates a single polynomial multiplied by a monomial. To solve this, distribute \( 2x^2 \) across each term in \( 3x^2 + 4x - 7 \), resulting in the multiplications: \( 3x^2 \times 2x^2 = 6x^4 \), \( 4x \times 2x^2 = 8x^3 \), and \( -7 \times 2x^2 = -14x^2 \).
Once the multiplication of each pair of terms is complete, sum the resulting expressions to form the final polynomial: \( 6x^4 + 8x^3 - 14x^2 \). This step is essential for combining like terms and achieving a simplified polynomial expression.