Problem 58 Find each product. \(-5 r(r+1)... [FREE SOLUTION] (2024)

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Polynomial expansion involves expanding expressions of polynomials, often through multiplication of binomials or higher-order terms. In our exercise, we need to expand (r+1)^3 to simplify the expression.
This involves breaking down the cubic binomial and multiplying it out step-by-step.
Here’s how you go from (r+1)^3:

• First, rewrite (r+1)^3 as (r+1)(r+1)(r+1).
• Next, multiply two of the binomials: (r+1)(r+1) to get r^2 + 2r + 1.
• Finally, multiply this result by (r+1) once more to obtain r^3 + 3r^2 + 3r + 1.

This complete expansion process breaks down a higher-order polynomial into simpler terms, which can then be further manipulated in our calculations. Expansion follows systematic steps ensuring each term interrelates correctly as per algebraic multiplication principles.

Algebraic multiplication is essentially multiplying algebraic expressions, terms, or polynomials systematically. This ensures accuracy and simplification of complex expressions.
After our polynomial expansion, we received an expression r^3 + 3r^2 + 3r + 1. Now we need to multiply this result by -5r.Apply distribution: (-5r) * (r^3 + 3r^2 + 3r + 1). Here’s the breakdown:

• Multiply -5r by each term individually:
• (-5r)(r^3) = -5r^4
• (-5r)(3r^2) = -15r^3
• (-5r)(3r) = -15r^2
• (-5r)(1) = -5r

Combining these products gives us the final expression: -5r^4 - 15r^3 - 15r^2 - 5r.
This ensures each term is correctly multiplied and combined, providing the simplified result of our initial complex term.

A binomial is an algebraic expression containing two different terms connected by a plus or minus sign. Understanding binomials is essential because they are the building blocks of polynomial expressions and are frequently expanded or multiplied in algebra. In our exercise, we started with a binomial (r+1) which was raised to the third power, making (r+1)^3. To simplify, we broke it down and treated it as (r+1)(r+1)(r+1).

• By knowing binomials, we were able to systematically expand this expression step-by-step:
• (r+1)^2 became r^2 + 2r + 1, and
• (r^2 + 2r + 1)(r+1) expanded into r^3 + 3r^2 + 3r + 1.

Grasping the properties and expansion methods of binomials allows us to handle more complex problems by breaking them down into these simpler components. This step ensures confidence with larger algebraic manipulations.