Questions and Answers
What is the placeholder needed when dividing $x^2 - 1$ by $x - 3$?
What is $(x - 3)^3$ expanded?
x^3 - 9x^2 + 27x - 27
What is the expanded form of $3x^2(x + 1)$?
3x^3 + 3x^2
What is one of the factored forms of $x^3 - 9x^2 + 27x - 27$?
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The expression $x^2 - 5$ is prime.
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When dividing a polynomial by a monomial, each term of the polynomial is divided by the monomial.
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What is the result of expanding $(4c + 5m)(4c - 5m)$?
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What is $(2y + 3)^2$ in expanded form?
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What is the result of expanding $(w + 12)(w - 12)$?
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What do you get when you factor $y^2 - 2y - 4$?
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Study Notes
Polynomial Division and Properties
- When dividing (x^2 - 1) by (x - 3), a placeholder of (-3x) is necessary for proper alignment.
- Dividing a polynomial by a monomial involves dividing each term of the polynomial individually by the monomial.
Factored Forms and Expansion
- The expression (x^3 - 9x^2 + 27x - 27) can be factored as ((x - 3)^3).
- Expansion of ((2y + 3)^2) results in (4y^2 + 12y + 9).
- The product ((4c + 5m)(4c - 5m)) simplifies to (16c^2 - 25m^2).
- The difference of squares formula is applied to ((w + 12)(w - 12)), resulting in (w^2 - 144).
Prime Expressions
- The expression (x^2 - 5) is considered prime, indicating it cannot be factored over the integers.
Polynomial Simplification
- The expression (y^2 - 2y - 4) can be rewritten as (\frac{y^3 - y^2 - 6y - 4}{y + 1}) showing the polynomial in terms of another variable.
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