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Questions and Answers
What is the greatest common factor (GCF) of the polynomial $12x^3 + 18x^2 - 24x$?
What is the greatest common factor (GCF) of the polynomial $12x^3 + 18x^2 - 24x$?
- $6x^3$
- $24x^3$
- $12x$
- $6x$ (correct)
Which expression represents the factored form of $6x^2 + 9x$ using the greatest common factor?
Which expression represents the factored form of $6x^2 + 9x$ using the greatest common factor?
- $3(2x^2 + 3x)$
- $x(6x + 9)$
- $3x(2x + 9)$
- $3x(2x + 3)$ (correct)
Factor the following expression by grouping: $3x^2 + 6x + 4x + 8$.
Factor the following expression by grouping: $3x^2 + 6x + 4x + 8$.
- $(3x + 6)(x + 8)$
- $(3x + 8)(x + 1)$
- $(3x + 4)(x + 2)$ (correct)
- $(3x + 2)(x + 4)$
What are the factors of the quadratic trinomial $x^2 - 5x + 6$?
What are the factors of the quadratic trinomial $x^2 - 5x + 6$?
Which of the following quadratic trinomials is a perfect square trinomial?
Which of the following quadratic trinomials is a perfect square trinomial?
Factor the perfect square trinomial: $9x^2 - 12x + 4$.
Factor the perfect square trinomial: $9x^2 - 12x + 4$.
What is the factored form of the difference of squares: $16x^2 - 25$?
What is the factored form of the difference of squares: $16x^2 - 25$?
Which of the following expressions is a difference of squares?
Which of the following expressions is a difference of squares?
Factor the expression completely: $2x^3 - 8x$.
Factor the expression completely: $2x^3 - 8x$.
Given the polynomial $x^3 + 5x^2 - 2x - 10$, which factoring method would be most appropriate to begin with?
Given the polynomial $x^3 + 5x^2 - 2x - 10$, which factoring method would be most appropriate to begin with?
Flashcards
What is Factorization?
What is Factorization?
Breaking down a polynomial into simpler polynomials or factors.
What is the Greatest Common Factor (GCF)?
What is the Greatest Common Factor (GCF)?
The largest factor that divides all terms of the polynomial.
What is Factoring by Grouping?
What is Factoring by Grouping?
A method used for polynomials with four terms, pairing terms and finding common factors.
What are Quadratic Trinomials?
What are Quadratic Trinomials?
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What are Perfect Square Trinomials?
What are Perfect Square Trinomials?
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What is a Difference of Squares?
What is a Difference of Squares?
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Study Notes
- Factorization decomposes a polynomial into simpler polynomials or factors.
Common Factor
- The greatest common factor (GCF) among all polynomial terms must be identified first.
- Factoring out the GCF from each term simplifies the polynomial.
- For example: the GCF in (4x + 6) is 2, leading to a factored form of (2(2x + 3)).
Factoring by Grouping
- Polynomials with four terms are sometimes factorable by grouping.
- Terms are paired and common factors identified within each pair via grouping.
- After each pair is factored separately, a common binomial factor is factored out from the entire expression, resulting in the final factored form.
Quadratic Trinomials
- Quadratic trinomials, in the form (ax^2 + bx + c), can often be factored into two binomials.
- Two numbers that multiply to (ac) and add up to (b) must be found.
- These numbers decompose the middle term, enabling factoring by grouping.
Perfect Square Trinomials
- Perfect square trinomials, like (a^2 + 2ab + b^2) or (a^2 - 2ab + b^2), have specific factorization patterns.
- (a^2 + 2ab + b^2) factors into ((a + b)^2).
- (a^2 - 2ab + b^2) factors into ((a - b)^2).
Difference of Squares
- A difference of squares, in the form (a^2 - b^2), factors into ((a + b)(a - b)).
- Recognizing this pattern allows for quick factorization.
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