Exponents and Polynomials Checkpoint
A free Algebra I lesson from the “Exponents and Polynomials” unit, with a worked example and practice problems including step-by-step solutions.
This checkpoint blends exponent rules with polynomial operations and factoring. The work should feel like one connected algebra toolkit: simplify, combine, distribute, and factor.
What you'll learn
- Apply exponent rules
- Operate with polynomials
- Factor common patterns
Why it matters: Polynomial tools help rewrite expressions into forms that are easier to calculate, compare, graph, or solve in design and modeling problems.
Worked example
Problem. Simplify 2x(3x + 4) + x^2.
- Distribute 2x to get 6x^2 + 8x.
- Add the remaining x^2 term.
- 6x^2 + x^2 = 7x^2, so the result is 7x^2 + 8x.
Answer: 7x^2 + 8x
Practice problems
1. Simplify x^5 times x^3.
Show solution
- Exponent Rules: First identify exactly what the question is asking: Simplify x^5 times x^3.
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- Same base multiplication means add exponents.
- Check the result by substituting or estimating: the response should match x^8 and make sense in the original problem.
Answer: x^8
2. Simplify y^9 divided by y^4.
Show solution
- Exponent Rules: First identify exactly what the question is asking: Simplify y^9 divided by y^4.
- 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.
- Check the result by substituting or estimating: the response should match y^5 and make sense in the original problem.
Answer: y^5
3. Simplify (a^3)^2.
Show solution
- Exponent Rules: First identify exactly what the question is asking: Simplify (a^3)^2.
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- Power to a power means multiply exponents.
- Check the result by substituting or estimating: the response should match a^6 and make sense in the original problem.
Answer: a^6
4. Simplify (4x + 3) + (2x - 9).
Show solution
- Polynomial Addition: First identify exactly what the question is asking: Simplify (4x + 3) + (2x - 9).
- For polynomial work, use degree, leading terms, and like terms to keep the expression organized.
- Combine x terms and constants.
- Check the result by substituting or estimating: the response should match 6x - 6 and make sense in the original problem.
Answer: 6x - 6
5. Simplify (7x - 1) - (3x + 5).
Show solution
- Polynomial Subtraction: First identify exactly what the question is asking: Simplify (7x - 1) - (3x + 5).
- For polynomial work, use degree, leading terms, and like terms to keep the expression organized.
- Distribute the subtraction.
- 7x - 1 - 3x - 5 = 4x - 6.
- Check the result by substituting or estimating: the response should match 4x - 6 and make sense in the original problem.
Answer: 4x - 6
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