Problem 56 - Algebra 1: Concepts and Skills

Question

Find the product. $$ 3 q\left(q^{3}-5 q^{2}+6\right) $$

Step-by-Step Solution

Answer

The product is \(3q^4 - 15q^3 + 18q\)

Step 1: Apply the Distributive Property

Applying the distributive law, which states that 'a(b + c + d) = ab + ac + ad', to the expression \(3q (q^3 - 5q^2 + 6)\), results in \(3q \cdot q^3\), \(3q \cdot -5q^2\), and \(3q \cdot 6\).

Step 2: Perform the Multiplication

Perform the multiplication within each calculated result from the first step: \(3q \cdot q^3\) equals \(3q^4\), \(3q \cdot -5q^2\) equals \(-15q^3\) , and \(3q \cdot 6\) equals \(18q\).

Step 3: Write Down the Final Answer

Combine the results from step 2 to write down the final answer. The product of distribution is \(3q^4 - 15q^3 + 18q\).

Key Concepts

Polynomial Multiplication

Polynomial multiplication is a process used to expand expressions where two polynomials are multiplied together. In the given exercise, we multiplied a monomial by a trinomial. This involves distributing the monomial across each term of the polynomial. The distributive property allows us to multiply each term in one polynomial by each term in the other polynomial. This step-by-step approach prevents mistakes and simplifies the expression into manageable parts.

When multiplying polynomials, each term of one polynomial is multiplied by every term of the other polynomial. After completing all multiplications, like terms are combined by adding or subtracting them to form a simplified polynomial.

• Start by distributing the first term.
• Continue distributing each subsequent term.
• Finally, combine like terms to simplify the expression further.

This systematic method ensures the multiplication is conducted accurately.

Monomial

A monomial is a single term expression consisting of a constant, a variable or the product of constants and variables. In this exercise, the monomial is \(3q\). It's important to understand the role of the monomial as it is multiplied with each term of the polynomial separately.

Monomials are generally easier to handle, but their multiplication can become complex due to the presence of variables and exponents. Here, the monomial \(3q\) is multiplied by each term in the trinomial \((q^3 - 5q^2 + 6)\).

• Multiply coefficients of the monomial and polynomial terms.
• Add exponents of like bases during multiplication.

This simplifies the multiplication process and makes it straightforward to combine or reduce terms.

Exponent Rules for Multiplication

The rules for multiplying exponents play a crucial role when working with polynomials. When you have terms with the same base, their exponents are added during multiplication. This foundational rule helps simplify expressions which involve variables raised to a power.

In the exercise, the monomial \(3q\) is multiplied by terms like \(q^3\) and \(-5q^2\). The multiplication of \(q\) and \(q^3\) involves adding the exponents of \(q\), resulting in \(q^4\). Similarly, \(q\) and \(q^2\) yield \(q^3\) after multiplication.

• Identify terms with like bases.
• Add their exponents when multiplying.
• Ensure the bases remain the same in the product.

These rules ensure terms are simplified correctly, resulting in a cleaner, more concise polynomial.