91Ó°ÊÓ

The distributive property is a fundamental principle in algebra that allows us to multiply a single term by each term within a parenthesis. This property states that for all real numbers a, b, and c: \(a(b + c) = ab + ac\). In the context of our exercise, we apply the distributive property twice. First, we distribute each term in the binomial \(x+1\) to each term in the polynomial \(x^2 - 3x - 21\). The distributive property helps to expand expressions and make multiplication straightforward.

For example, distributing the first term in our problem involves multiplying 1 by each term in the polynomial: \(1 \cdot x^2 - 1 \cdot 3x - 1 \cdot 21 = x^2 - 3x - 21\). This shows how each term is handled individually, simplifying the multiplication process.

Combining like terms is essential in simplifying algebraic expressions. Like terms are terms that have the same variables raised to the same power. For example, \(3x^2\) and \(-2x^2\) are like terms, but \(x\) and \(x^2\) are not.

In our exercise, after using the distributive property, we have two separate sets of terms from distributing 1 and x: \(x^2 - 3x - 21\) and \(x^3 - 3x^2 - 21x\). To combine like terms:

• Combine \(x^2\) and \(-3x^2\)
• Combine \(-3x\) and \(-21x\)
• Finally, add any constants together.

This results in bringing together terms in a simpler, single expression: \(x^3 - 2x^2 - 24x - 21\).

A binomial is a polynomial with exactly two terms. In our exercise, the binomial is \(x+1\). Binomials are often utilized in algebraic multiplication, requiring an understanding of both the distributive property and how to handle multiple terms.

When we multiply a binomial by a polynomial, each part of the binomial needs to be considered. First, take the \('+1'\) part and distribute it across each term of the polynomial. Then, take the \('x'\) part and do the same.

This multiplication-by-parts ensures every term from the binomial gets multiplied by every term in the polynomial, ensuring we capture the full expanded expression.

Our practice problem shows this clearly:

• distribute \('1'\) and then \('x'\)
• combine like terms from resulting expressions
• simplify to get the final result

Multiplying binomials by polynomials might look challenging at first, but by breaking it down into steps, it becomes manageable.