Guided Practice
Work through polynomial and radical expression problems with step-by-step guidance to reinforce your understanding of Algebra II concepts.
Learn
This guided practice lesson helps you apply the concepts from Polynomial Operations and Radical Expressions. Each problem includes hints and step-by-step solutions to guide your learning.
Key Concepts Review
- Polynomial Addition/Subtraction: Combine like terms by adding or subtracting coefficients
- Polynomial Multiplication: Use the distributive property (FOIL for binomials)
- Radical Simplification: Factor out perfect squares from under the radical
- Rationalizing Denominators: Multiply by the conjugate to eliminate radicals in denominators
Worked Examples
Study these examples carefully before attempting the practice problems.
Example 1: Polynomial Multiplication
Problem: Expand (2x + 3)(x - 5)
Step 1: Apply FOIL (First, Outer, Inner, Last)
First: 2x * x = 2x2
Outer: 2x * (-5) = -10x
Inner: 3 * x = 3x
Last: 3 * (-5) = -15
Step 2: Combine like terms
2x2 - 10x + 3x - 15 = 2x2 - 7x - 15
Example 2: Simplifying Radicals
Problem: Simplify sqrt(72)
Step 1: Factor 72 to find perfect square factors
72 = 36 * 2
Step 2: Apply the product rule for radicals
sqrt(72) = sqrt(36 * 2) = sqrt(36) * sqrt(2) = 6sqrt(2)
Example 3: Rationalizing Denominators
Problem: Rationalize 5 / sqrt(3)
Step 1: Multiply numerator and denominator by sqrt(3)
(5 / sqrt(3)) * (sqrt(3) / sqrt(3))
Step 2: Simplify
= 5sqrt(3) / 3 = (5sqrt(3))/3
Practice Problems
Complete these 10 practice problems. Each problem builds on the concepts you've learned.
Problem 1
Expand: (x + 4)(x + 7)
Show Hint
Use FOIL: First terms, Outer terms, Inner terms, Last terms.
Show Answer
x2 + 11x + 28
Problem 2
Expand: (3x - 2)(2x + 5)
Show Hint
Apply FOIL carefully with the negative sign.
Show Answer
6x2 + 11x - 10
Problem 3
Simplify: (4x2 + 3x - 7) + (2x2 - 5x + 9)
Show Hint
Combine like terms: x2 terms, x terms, and constants separately.
Show Answer
6x2 - 2x + 2
Problem 4
Simplify: sqrt(50)
Show Hint
Factor 50 = 25 * 2, where 25 is a perfect square.
Show Answer
5sqrt(2)
Problem 5
Simplify: sqrt(128)
Show Hint
128 = 64 * 2. What is sqrt(64)?
Show Answer
8sqrt(2)
Problem 6
Rationalize: 3 / sqrt(5)
Show Hint
Multiply top and bottom by sqrt(5).
Show Answer
(3sqrt(5))/5
Problem 7
Expand: (x - 3)2
Show Hint
Use the pattern (a - b)2 = a2 - 2ab + b2
Show Answer
x2 - 6x + 9
Problem 8
Simplify: sqrt(75) + sqrt(27)
Show Hint
Simplify each radical first: 75 = 25*3 and 27 = 9*3
Show Answer
5sqrt(3) + 3sqrt(3) = 8sqrt(3)
Problem 9
Subtract: (5x2 - 2x + 8) - (3x2 + 4x - 2)
Show Hint
Distribute the negative sign to all terms in the second polynomial.
Show Answer
2x2 - 6x + 10
Problem 10
Rationalize: 4 / (sqrt(2) + 1)
Show Hint
Multiply by the conjugate (sqrt(2) - 1) / (sqrt(2) - 1)
Show Answer
4(sqrt(2) - 1) / (2 - 1) = 4sqrt(2) - 4
Check Your Understanding
Answer these questions to verify you've mastered the guided practice concepts.
1. When multiplying (a + b)(a - b), what pattern do you get?
Show Answer
The difference of squares: a2 - b2
2. Why do we rationalize denominators?
Show Answer
To eliminate radicals from the denominator, making expressions easier to work with and compare.
Next Steps
- Review any problems where you needed hints
- Practice similar problems until you can solve them without hints
- Move on to Word Problems to apply these skills in context