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
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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.