Name: Unit 5: Polynomial Functions Homework
1: Monomials & Polynomials
Date:
Bell:
Directions: Simplify the monomials below. Final answers should contain positive exponents only.
Subtract -3n2 from -7n2
(-8xy) – (2xy) + 7x0y5
(-4d2b2) (3a2b)
507d2 (-cd2)
15x1y – 6x1y + 18w0x0 + 14wx5
(3xy)2
The Correct Answer and Explanation is:
Let’s go step by step to solve and explain the last expression:
Expression:
(3xy)²
Step-by-step Solution:
This is an expression involving exponents and multiplication of variables inside parentheses.
Step 1: Apply the exponent rule: (ab)ⁿ = aⁿ * bⁿ
So:
[
(3xy)^2 = 3^2 \cdot x^2 \cdot y^2
]
Step 2: Simplify the powers:
- (3^2 = 9)
- (x^2 = x^2)
- (y^2 = y^2)
Final Answer:
[
\boxed{9x^2y^2}
]
300-Word Explanation:
The expression ((3xy)^2) is a monomial raised to a power. When simplifying such expressions, we use the Power of a Product Rule from exponent laws. This rule states that when an entire product inside parentheses is raised to an exponent, we must apply that exponent to each factor individually. In mathematical terms, this rule is:
[
(ab)^n = a^n \cdot b^n
]
In our problem, the entire product (3xy) is being squared, or raised to the second power. According to the rule above, this means:
[
(3xy)^2 = 3^2 \cdot x^2 \cdot y^2
]
We then compute each individual exponent:
- (3^2 = 9), because 3 times 3 is 9.
- (x^2 = x \cdot x), which simply represents x multiplied by itself.
- (y^2 = y \cdot y), again showing the variable y multiplied by itself.
Putting it all together, the expression simplifies to (9x^2y^2). This is a fully simplified monomial, with all variables having positive exponents, which is a requirement of the directions.
Understanding this process is essential when working with polynomials and algebraic expressions. It demonstrates how exponents distribute over multiplication and helps build the foundation for handling more complex expressions like binomials and polynomials raised to powers. Correct simplification ensures clarity and accuracy, especially when combining like terms or solving polynomial equations.