The rules for solving polynomials involve finding the values of the variable that make the polynomial equal to zero. Here's a breakdown of the process:
Steps to Solve Polynomials
Solving polynomials generally involves the following steps:
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Simplify the Polynomial: Combine like terms and arrange the polynomial in descending order of exponents. This makes it easier to identify the degree and coefficients.
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Factorization (if possible): Try to factor the polynomial. This can be achieved using various techniques:
- Greatest Common Factor (GCF): Look for a common factor among all terms.
- Difference of Squares: For polynomials in the form a² - b², factor them as (a + b)(a - b).
- Perfect Square Trinomial: Recognize and factor patterns like a² + 2ab + b² as (a + b)² or a² - 2ab + b² as (a - b)².
- Grouping: When the polynomial has four or more terms, group terms to find common factors.
- Trial and Error (for quadratics): For quadratic trinomials (ax² + bx + c), try different combinations of factors.
- Factoring by the AC method. For quadratic trinomials (ax² + bx + c), multiply a and c, then find two factors of ac that sum to b.
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Division (long or synthetic): If factorization is not immediately obvious, use long division or synthetic division to divide the polynomial by a known factor, like (x - a) where 'a' is a root, which could be found by using techniques like the rational root theorem. According to reference #1: "Divide the polynomial expression using long division or synthetic division." This can help to simplify the polynomial and make it easier to factor.
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Express as Linear Factors: After factorization or division, express the polynomial as a product of its linear factors. Reference #2 instructs to "Express the polynomial as a product of its' linear factors."
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Apply the Null Factor Law: Set each factor equal to zero and solve for the variable. The Null Factor Law (or zero product property) states that if the product of several factors is zero, then at least one of the factors must be zero. According to reference #2: "Use the null factor law to find the solution(s)."
Examples
Example 1: Factoring a Quadratic
- Solve: x² - 5x + 6 = 0
- Factor: (x - 2)(x - 3) = 0
- Apply Null Factor Law: x - 2 = 0 or x - 3 = 0
- Solutions: x = 2 or x = 3
Example 2: Factoring with a Common Factor
- Solve: 2x³ + 6x² = 0
- Factor out GCF: 2x²(x + 3) = 0
- Apply Null Factor Law: 2x²=0 or x + 3 = 0
- Solutions: x = 0 or x = -3
Example 3: Using Division
- Solve: x³ - 6x² + 11x - 6 = 0, given one root is x=1
- Divide the polynomial by (x-1) using synthetic division or long division to yield: (x-1)(x² - 5x + 6) = 0
- Factor: (x - 1)(x - 2)(x - 3) = 0
- Apply Null Factor Law: x - 1 = 0, x - 2 = 0, or x - 3 = 0
- Solutions: x = 1, x = 2, or x = 3
Key Points
- Degree: The highest power of the variable in the polynomial determines the maximum number of roots the polynomial can have.
- Real and Complex Roots: Polynomials can have real or complex (imaginary) roots.
- Rational Root Theorem: Can help find possible rational roots.
- Numerical Methods: Sometimes numerical methods are needed to find approximate solutions when factorization is difficult or impossible.
By following these rules and employing different factorization and division techniques, you can solve a wide variety of polynomial equations.
