In Exercises 9–22, multiply the monomial and the polynomial.5x³ (2x⁵−4x²+9)

Verified step by step guidance

1. Identify the expression to be multiplied: \(5x^3\) is the monomial and \((2x^5 - 4x^2 + 9)\) is the polynomial.
2. Apply the distributive property: Multiply the monomial \(5x^3\) by each term in the polynomial separately.
3. Multiply \(5x^3\) by the first term \(2x^5\): Use the rule \(a^m \cdot a^n = a^{m+n}\) to combine the exponents.
4. Multiply \(5x^3\) by the second term \(-4x^2\): Again, use the exponent rule to combine the exponents.
5. Multiply \(5x^3\) by the constant term \(9\): Remember that multiplying by a constant does not change the variable part.

Key Concepts

Monomials

A monomial is a single term algebraic expression that consists of a coefficient and one or more variables raised to non-negative integer powers. For example, in the expression 5x³, 5 is the coefficient and x is the variable raised to the power of 3. Understanding monomials is essential for performing operations like multiplication with polynomials.

Polynomials

A polynomial is an algebraic expression that consists of multiple terms, each of which is a monomial. Polynomials can be classified by their degree, which is the highest power of the variable in the expression. In the given example, 2x⁵ - 4x² + 9 is a polynomial with three terms, and recognizing its structure is crucial for correctly applying multiplication with a monomial.

Distributive Property

The distributive property states that a(b + c) = ab + ac, allowing us to multiply a single term by each term in a polynomial. This property is fundamental when multiplying a monomial by a polynomial, as it ensures that each term in the polynomial is multiplied by the monomial separately, leading to the correct expansion of the expression.