Questions and Answers
What does FOIL stand for?
First, Outer, Inner, Last
What is the equation to factor binomials?
x^2 + bx + c
How do you know which factor of c should be positive and which should be negative when factoring a trinomial?
If the c term is a positive number, then the factors of c will both be positive or both be negative.
How do you decide to factor a positive or negative GCF?
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What is the Zero Product Property?
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How would you solve x^2 + 15x = -54?
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How do you solve x^2 + bx + c = 0 by factoring?
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What are the dimensions of the room if the area is 396 square feet with length (x + 3) and width (x + 7) feet?
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What is the first step when you see the equation 4x^2 + 26x + 42?
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What is the next step after factoring out the GCF?
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What is the next step after listing factor pairs of 2?
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What is the next step after listing factor pairs of 21?
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What is the next step after making the combinations table?
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What is the last step in the factorization?
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Using the above steps, how would you factor 5x^2 + 8x + 4?
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What are the dimensions of a rectangular closet before and after construction?
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How can you use special products to aid in solving quadratic equations by factoring?
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If the middle term in x^2 + 6x + 9 was changed to 10x, how does this affect factoring?
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How can a perfect-square trinomial be represented algebraically?
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How do you factor the perfect square trinomial 4x^3 - 24x^2 + 36x?
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How do you factor each difference of squares: x^2 - 144?
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How can equations with special factors be solved?
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Solve the equation with special factors: 25x^2 - 10x + 1 = 0.
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Explain the process of solving 10.A volleyball player sets the ball in the air with height h = -16t^2 + 12t + 6.
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How would you go about factoring a^2 - 2ab + b^2 - 1?
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Solve using special factors: 25x^2 + 20x + 4 = 0.
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When has Jivesh's rocket descended 1000 centimeters?
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Explain how to fully factor the expression mentioned.
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Study Notes
FOIL Method
- FOIL stands for First, Outer, Inner, Last; a method for multiplying two binomials.
Factoring Binomials
- To factor binomials, use the equation x² + bx + c.
Factors of a Negative c Term
- When factoring x² + bx + c with a negative c, one factor is positive and the other is negative; both factors of c are either positive or negative if c is positive.
Factoring GCF
- When deciding between factoring a positive or negative GCF, check the signs of the second and fourth terms: same signs indicate a positive GCF, different signs indicate a negative GCF.
Zero Product Property
- The Zero Product Property states that if AB = 0, then A = 0 or B = 0.
Solving Quadratic Equations
- To solve x² + 15x = -54, rearrange to standard form and factor or use the quadratic formula.
Factoring Quadratics
- To solve x² + bx + c = 0 by factoring, identify factor pairs that sum to b and use them to form binomials.
Area Problem
- For a room with an area of 396 square feet, length (x + 3) and width (x + 7), set up the equation (x + 3)(x + 7) = 396 to find dimensions.
Common Factor
- In 4x² + 26x + 42, start by factoring out the greatest common factor, which is 2.
Factor Pairs
- List factor pairs of the resulting numbers to find combinations that fulfill the equation's requirements.
Factor Sum Table
- Create a table with factor combinations of a and c to find which SUMS up to b using the outer and inner products.
Identifying Binomial Factors
- Copy pairs of factors with an outer + inner sum of 13 into binomial factors for completion.
Completing Factorization
- Replace the common factor in the original coefficients to finalize the quadratic's factorization.
Closet Area Problem
- For a closet with area (x² + 3x + 2) and enlarged area (x² + 8x + 15), determine dimensions and changes.
Special Products and Factoring
- Special products like perfect-square trinomials can simplify the factorization process.
Impact of Middle Term Change
- Altering the middle term (e.g., from 6x to 10x) affects the identification of factors and their pairs.
Perfect-Square Trinomials
- Represent perfect-square trinomials in the forms a² + 2ab + b² or a² - 2ab + b² for easier factoring.
Factoring Particular Trinomials
- To factor trinomials like 4x³ - 24x² + 36x, apply techniques specific for polynomials of higher degree.
Difference of Squares
- Apply the difference of squares method to factor expressions like x² - 144.
Zero Product Property Application
- Use the Zero Product Property for solving equations with special factors by setting each factor to zero.
Solving Quadratic Equations
- Solve 25x² - 10x + 1 = 0 using factoring or the quadratic formula.
Volleyball Height Problem
- For the height equation h = -16t² + 12t + 6, find when it reaches 8 feet and identify reasonable solutions.
Factoring Trinomials with Constant
- When factoring expressions like a² - 2ab + b² - 1, employ techniques for grouping or completing the square.
Special Factors Application
- Solve equations involving special factors, exemplified by 25x² + 20x + 4 = 0, by determining if they can be factored easily.
Rocket Descent Model
- Jivesh's rocket descent modeled by h = -490t²; determine when it descends 1000 centimeters using the quadratic equation.
General Factoring Approach
- A methodical approach for factoring includes rearranging, simplifying, and applying techniques based on product properties.
Studying That Suits You
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Description
Test your understanding of algebra concepts in Chapter 21 with these flashcards. This quiz covers essential terms and equations, including the FOIL method and factoring techniques for binomials and trinomials. Challenge yourself to master the material and solidify your knowledge in algebra.
