Exponents and Polynomials
A free Algebra I lesson from the “Exponents and Polynomials” unit, with a worked example and practice problems including step-by-step solutions.
Exponent rules help simplify repeated multiplication. Polynomials are expressions made of terms with whole-number exponents. Like terms can be combined across polynomials.
What you'll learn
- Apply exponent rules
- Classify polynomial expressions
- Add and subtract polynomials
Why it matters: Exponents and polynomials show up in area formulas, compact growth patterns, and expressions that describe repeated structure.
Worked example
Problem. Simplify x^3 times x^4.
- The bases are the same.
- When multiplying powers with the same base, add exponents.
- 3 + 4 = 7, so the result is x^7.
Answer: x^7
Practice problems
1. Simplify x^2 times x^5.
Show solution
- Warm-up: First identify exactly what the question is asking: Simplify x^2 times x^5.
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- Same base means add exponents.
- 2 + 5 = 7.
- Check the result by substituting or estimating: the response should match x^7 and make sense in the original problem.
Answer: x^7
2. Simplify a^6 divided by a^2.
Show solution
- Warm-up: First identify exactly what the question is asking: Simplify a^6 divided by a^2.
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- Same base division means subtract exponents.
- 6 - 2 = 4.
- Check the result by substituting or estimating: the response should match a^4 and make sense in the original problem.
Answer: a^4
3. How many terms are in 3x^2 + 5x - 8?
Choices: 1 · 2 · 3 · 4
Show solution
- Warm-up: First identify exactly what the question is asking: How many terms are in 3x^2 + 5x - 8?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Terms are separated by plus or minus signs.
- There are 3 terms.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: 3
4. Simplify (2x + 5) + (3x - 1).
Show solution
- Core Practice: First identify exactly what the question is asking: Simplify (2x + 5) + (3x - 1).
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- Combine x terms: 2x + 3x = 5x.
- Combine constants: 5 - 1 = 4.
- Check the result by substituting or estimating: the response should match 5x + 4 and make sense in the original problem.
Answer: 5x + 4
5. Simplify (7x - 4) - (2x + 3).
Show solution
- Core Practice: First identify exactly what the question is asking: Simplify (7x - 4) - (2x + 3).
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- Distribute the subtraction: 7x - 4 - 2x - 3.
- Combine to get 5x - 7.
- Check the result by substituting or estimating: the response should match 5x - 7 and make sense in the original problem.
Answer: 5x - 7
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