Problem 77 - Elementary Algebra

Question

\(8 k\left(2 k^{2}-k+5\right)\)

Step-by-Step Solution

Answer

16k^3 - 8k^2 + 40k

Step 1: Distribute the 8k

Apply the distributive property to multiply each term inside the parentheses by 8k. The expression is: \[8k \times (2k^2) + 8k \times (-k) + 8k \times 5\]

Step 2: Multiply each term

Perform the multiplications: \[8k \times 2k^2 = 16k^3\], \[8k \times -k = -8k^2\], and \[8k \times 5 = 40k\]

Step 3: Write the final simplified expression

Combine the simplified terms to get the final answer: \[16k^3 - 8k^2 + 40k\]

Key Concepts

Polynomial Multiplication

Polynomial multiplication refers to multiplying two or more polynomials together. It involves using distributive property. For example, consider multiplying a polynomial by a monomial, as in the exercise: When multiplying a polynomial like \(8k(2k^2 - k + 5)\), distribute the monomial \(8k\) to each term inside the parentheses:

• Multiply \(8k \) by \2k^2\
• Multiply \(8k \) by \ -k\
• Multiply \(8k \) by \5\

This systematic distribution allows for more manageable calculations. Hence \(8k \) multiplied by each term inside gives: \[8k \cdot 2k^2 + 8k \cdot -k + 8k \cdot 5 = 16k^3 - 8k^2 + 40k\]

Simplifying Expressions

Simplifying expressions involves performing all possible arithmetic operations and combining like terms to reduce the expression to its simplest form. It makes working with complex expressions easier and clear.

To simplify our example:

• First, distribute \(8k\) to each term inside the parentheses. This removes the parentheses.
• Next, carry out the multiplications: \8k \times 2k^2\ becomes \16k^3\, \8k \times -k\ becomes \-8k^2\, and \8k \times 5\ becomes \40k\.
• Finally, after performing all multiplications, you combine the results: \16k^3\, \-8k^2\, and \40k\.

This gives the final simplified form: \16k^3 - 8k^2 + 40k\.

Algebraic Expressions

An algebraic expression is composed of variables, coefficients, and constants combined using arithmetic operations such as addition, subtraction, multiplication, and division.

For instance, in the expression \(8k(2k^2 - k + 5)\)

Importance of Algebraic Expressions

They are fundamental in algebra for representing real-world situations and solving problems. Simplify and manipulate expressions using properties like the distributive property to make solving equations easier and more understandable.