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Questions and Answers
What type of equation is represented by $(x  1)^2  1 + x^2 = (1  x)(x + 3)$?
Which equation is a cubic equation?
What is the highest degree of the polynomial in the equation $x^4 + 3x^3 + 8x^2 + 6x + 5 = 0$?
Which of the following equations is not a polynomial equation?
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What type of roots would you expect from the equation $x^4 + x^3 + x + 1 = 0$?
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In the equation $4x^3  10x^2 + 14x  5 = 0$, what can be inferred about its solutions?
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Which equation involves both cubic and linear terms?
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What defines a quartic equation?
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What is the degree of the polynomial in equation 1: $x^4  10x^2  x + 20 = 0$?
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In the equation $x^4  4x^2 + 12x  9 = 0$, which term does not affect the degree of the polynomial?
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Which equation represents a polynomial of degree 3?
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What is the correct factored form of the polynomial $x^6 + 3x^5  6x^4  21x^3  6x^2 + 3x + 1 = 0$?
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Which of the following equations contains only even powers of x?
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Which polynomial can be factored to include a common binomial factor?
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What is the outcome when setting the equation $2x^4  5x^3 + 6x^2  5x + 2 = 0$ to zero?
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Which of the following represents a polynomial identity?
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Study Notes
Solving Higher Degree Equations Using Factoring
 Many higher degree equations can be solved by factoring them into simpler expressions.
 This allows you to find solutions by setting each factor equal to zero and solving for x.
 The key is to identify patterns and use algebraic manipulations to factor the equation.
Common Techniques
 Grouping: Look for common factors within groups of terms.
 Difference of Squares: Recognize expressions in the form (a²  b²) which can be factored as (a + b)(a  b).
 Sum/Difference of Cubes: Identify expressions in the form (a³ + b³) or (a³  b³) which can be factored using specific formulas.
 Substitution: Introduce a new variable (e.g., let y = x²) to simplify the equation and make it easier to factor.
Examples

Example 1: x⁴  10x²  x + 20 = 0

Factoring by grouping:
 Group the first two terms and the last two terms:
 (x⁴  10x²)  (x  20) = 0
 Factor out common factors:
 x²(x²  10)  1(x  20) = 0
 Notice that (x²  10) and (x  20) can't be further factored.
 This equation does not factor easily, and may require other methods to solve.
 Group the first two terms and the last two terms:

Factoring by grouping:

Example 2: (x + 2)(x  3)(x + 4)(x  6) + 6x² = 0

Factoring a difference of squares:
 Notice that the first four terms represent a product of four factors, and the last term is a perfect square.
 By rearranging and factoring, it can be rewritten as:
 [(x + 2)(x  6)] [(x  3)(x + 4)] + 6x² = 0
 (x²  4x  12)(x² + x  12) + 6x² = 0
 Identify a difference of squares:
 (x²  4x  12 + √6x)(x²  4x  12  √6x) = 0
 Factor each quadratic expression separately:
 (x  2 + √6)(x  2 √6)(x  6)(x + 2) = 0
 Setting each factor equal to zero leads to the solutions for x.

Factoring a difference of squares:

Example 3: 3(x²  x + 1)²  2(x + 1)² = 5(x³ + 1)

Factoring a sum of cubes:
 Simplify and rewrite the expression as:
 3(x²  x + 1)²  2(x + 1)²  5(x + 1)(x²  x + 1) = 0
 Notice that (x²  x + 1) appears multiple times.
 Let y = (x²  x + 1):
 3y²  2(x + 1)²  5(x + 1)y = 0
 This is now a quadratic equation in y. Solve for y, and then substitute back to find the solutions for x.
 Simplify and rewrite the expression as:

Factoring a sum of cubes:
Important Notes
 Not all higher degree equations are solvable by factoring.
 Always try to simplify and rearrange terms before attempting to factor.
 Look for patterns and commonalities in the expression.
 Be aware of factoring methods specific to certain forms of equations.
 Practice with various examples to develop your problemsolving skills.
 Consider other methods, such as the rational root theorem or numerical methods, when factoring proves difficult.
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Description
Explore the techniques for solving higher degree equations through factoring. Learn to identify patterns, use algebraic manipulations, and apply common factoring methods such as grouping and the difference of squares. This quiz will help solidify your understanding of these essential algebraic concepts.