Problem 72 - Elementary Algebra

Question

\(-5 k^{2}(2 k-6)\)

Step-by-Step Solution

Answer

-10k^{3} + 30k^{2}

Step 1: Distribute the term outside the parentheses

Start by distributing -5k^{2} to each term inside the parentheses: -5k^{2}(2k) - 5k^{2}(-6)

Step 2: Distribute and simplify

Multiply -5k^{2} by 2k and -6 separately: (-5k^{2})(2k) = -10k^{3} and (-5k^{2})(-6) = 30k^{2}

Step 3: Combine the terms

Add the resulting terms together: -10k^{3} + 30k^{2}

Key Concepts

distributive property

The distributive property is a methodology often used in algebra. It helps simplify expressions and solve equations more efficiently. When you have a term outside parentheses, you distribute, or multiply, that term by each term inside the parentheses.
For example, in the expression \(-5k^{2}(2k-6)\), you distribute \(-5k^{2}\) to both \(2k\) and \(-6\):
\(-5k^{2}(2k) \) and \(-5k^{2}(-6)\).
This means \(-5k^2 \) multiplies each term inside the parentheses separately.
This property is essential when dealing with expressions involving multiple terms within parentheses.

multiplication of polynomials

The multiplication of polynomials is another crucial concept in algebra. This process involves multiplying each term in the first polynomial by each term in the second polynomial.
Looking at our example again, \(-5 k^{2}(2 k - 6)\), we need to multiply \(-5 k^{2}\) by each term inside the parentheses.
Step by step, we get:

• \(-5k^{2} \times 2k = -10k^{3}\)
• \(-5k^{2} \times -6 = 30k^{2}\)

This multiplication uses the distributive property, breaking down the expression into manageable parts.
By the end, you multiply the coefficients (numbers) and add the exponents when the bases (variables) are the same.

combining like terms

Combining like terms means simplifying an expression by adding or subtracting terms with the same variables and exponents.
After applying the distributive property and multiplying the polynomials, we get two terms: \(-10k^{3}\) and \(30k^{2}\).
In this case, there are no like terms to combine because \(-10k^{3}\) and \(30k^{2}\) have different exponents.
So the final simplified form of the expression is:
\(-10k^{3} + 30k^{2}\).
This concept is vital for simplifying polynomial expressions to their most reduced form.