Questions and Answers
What is the result of ((6x^2 - 2x - 28) / (2x + 4))?
3x - 7
Expand (3x^2(4x^2 - 5x + 7)).
12x^4 - 15x^3 + 21x^2
What is the result of ((4x - 5)(2x^2 + 3x - 6))?
8x^3 + 2x^2 - 39x + 30
A 9th degree polynomial with a negative leading coefficient rises as (x \to -\infty).
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Identify the degree, number of terms, and name of the polynomial: (3x^4 + x^2 - 5x + 9).
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A 10th degree polynomial with a negative leading coefficient falls as (x \to -\infty).
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What is the result of ((5x - 2 + y) - (-3y + 5x + 2))?
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A 3rd degree polynomial with a positive leading coefficient rises as (x \to -\infty).
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What is the result of (-4x + 7 - (5x - 3))?
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A 6th degree polynomial with a positive leading coefficient rises as (x \to -\infty).
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Factor (x^2 + 36).
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Factor (x^2 - 10x + 24) using the quadratic formula.
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Factor (x^2 - 81).
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Factor the polynomial (x^4 - 1 = 0).
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What is a polynomial that has roots 4 and 5i?
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Factor (x^2 + 3).
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Identify the degree, number of terms, and name of the polynomial: (6y^5 + 9y^2 - 3y + 8).
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Factor (x^2 - 1).
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Identify the degree, number of terms, and name of the polynomial: (5y - 10).
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Identify the degree, number of terms, and name of the polynomial: (7x^3 + 5^2 + 1).
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Study Notes
Polynomial Operations
- Simplifying the expression ((6x^2 - 2x - 28) / (2x + 4)) yields (3x - 7).
- The expression (3x^2(4x^2 - 5x + 7)) expands to (12x^4 - 15x^3 + 21x^2).
- The product of ((4x - 5)(2x^2 + 3x - 6)) results in (8x^3 + 2x^2 - 39x + 30).
Polynomial Characteristics
- A polynomial of degree 9 with a negative leading coefficient rises as (x \to -\infty) and falls as (x \to \infty).
- The polynomial (3x^4 + x^2 - 5x + 9) has degree 4, consists of 4 terms, and is classified as a 4th degree polynomial.
- A 10th degree polynomial with a negative leading coefficient falls for both (x \to -\infty) and (x \to \infty).
Simplifying Expressions
- The expression ((5x - 2 + y) - (-3y + 5x + 2)) simplifies to (10x - 2y).
- The simplification of (-4x + 7 - (5x - 3)) results in (-4x + 10).
Behavior of Polynomials
- A 3rd degree polynomial with a positive leading coefficient falls as (x \to -\infty) and rises as (x \to \infty).
- A 6th degree polynomial with a positive leading coefficient rises for both (x \to -\infty) and (x \to \infty).
Factoring Polynomials
- The polynomial (x^2 + 36) can be factored as ((x - 6i)(x + 6i)).
- The quadratic (x^2 - 10x + 24) factors to ((x - 6)(x - 4)).
- The expression (x^2 - 81) can be factored into ((x - 9)(x + 9)).
Finding Roots
- The equation (x^4 - 1 = 0) has roots of (±1, ±i).
- Given roots (4) and (5i), constructing a polynomial yields (x^3 - 4x^2 + 25x - 100).
- For (x^2 + 3), the roots are represented as ((x - i\sqrt{3})(x + i\sqrt{3})).
Degrees and Classifications
- The polynomial (6y^5 + 9y^2 - 3y + 8) has degree 5, includes 4 terms, and is categorized as a 5th degree polynomial.
- The polynomial (x^2 - 1) factors to ((x - \sqrt{1})(x + \sqrt{1})).
- The polynomial (5y - 10) has degree 1, consists of 2 terms, and is called a 1st degree binomial.
- The polynomial (7x^3 + 5^2 + 1) has degree 3, includes 3 terms, and is defined as a 3rd degree trinomial.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Prepare for your Algebra review with these flashcards focusing on Units 6 and 7. Each card presents a polynomial function and its properties, helping you understand the concepts of degrees, leading coefficients, and polynomial expressions. Perfect for students looking to strengthen their algebra skills.
