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Questions and Answers
What is the process of expressing a polynomial as a product of simpler polynomials called?
What is the process of expressing a polynomial as a product of simpler polynomials called?
Which type of factorization involves factoring a polynomial into factors with rational coefficients?
Which type of factorization involves factoring a polynomial into factors with rational coefficients?
What is the first step in the method of factoring out the greatest common factor (GCF) of a polynomial?
What is the first step in the method of factoring out the greatest common factor (GCF) of a polynomial?
What is the formula used in factoring quadratic polynomials?
What is the formula used in factoring quadratic polynomials?
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What is the result stated by the Fundamental Theorem of Algebra?
What is the result stated by the Fundamental Theorem of Algebra?
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What is the statement of the Factor Theorem?
What is the statement of the Factor Theorem?
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What method of factorization involves expressing the polynomial as a sum of simpler polynomials and then factoring each simpler polynomial?
What method of factorization involves expressing the polynomial as a sum of simpler polynomials and then factoring each simpler polynomial?
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What is the first step in the method of factoring by grouping?
What is the first step in the method of factoring by grouping?
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What is the purpose of factorization of polynomials?
What is the purpose of factorization of polynomials?
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Study Notes
Factorization of Polynomials
Definition
- Factorization of a polynomial is the process of expressing it as a product of simpler polynomials, called factors.
Types of Factorization
- Factorization over integers: Factoring a polynomial into factors with integer coefficients.
- Factorization over rational numbers: Factoring a polynomial into factors with rational coefficients.
- Factorization over real numbers: Factoring a polynomial into factors with real coefficients.
Methods of Factorization
-
Factoring out the greatest common factor (GCF):
- Find the GCF of all terms in the polynomial.
- Divide each term by the GCF.
- Write the result as a product of the GCF and the remaining polynomial.
-
Factoring by grouping:
- Group terms with common factors.
- Factor out the common factor from each group.
- Write the result as a product of the factors.
-
Factoring quadratic polynomials:
- Use the formula:
x^2 + bx + c = (x + d)(x + e). - Find the values of
dandethat satisfy the equation.
- Use the formula:
-
Factoring by decomposition:
- Express the polynomial as a sum of simpler polynomials.
- Factor each simpler polynomial.
- Write the result as a product of the factors.
Important Results
- Fundamental Theorem of Algebra: Every non-constant polynomial has at least one complex root.
-
Factor Theorem: If
ais a root of the polynomialf(x), then(x - a)is a factor off(x). -
Remainder Theorem: If
f(x)is divided by(x - a), the remainder isf(a).
Factorization of Polynomials
- Factorization is the process of expressing a polynomial as a product of simpler polynomials.
- Factors can be integers, rational numbers, or real numbers.
Types of Factorization
- Factorization over integers: factors have integer coefficients.
- Factorization over rational numbers: factors have rational coefficients.
- Factorization over real numbers: factors have real coefficients.
Methods of Factorization
- Factoring out the greatest common factor (GCF):
- Find the GCF of all terms in the polynomial.
- Divide each term by the GCF.
- Write the result as a product of the GCF and the remaining polynomial.
- Factoring by grouping:
- Group terms with common factors.
- Factor out the common factor from each group.
- Write the result as a product of the factors.
- Factoring quadratic polynomials:
- Use the formula:
x^2 + bx + c = (x + d)(x + e). - Find the values of
dandethat satisfy the equation.
- Use the formula:
- Factoring by decomposition:
- Express the polynomial as a sum of simpler polynomials.
- Factor each simpler polynomial.
- Write the result as a product of the factors.
Important Results
- Fundamental Theorem of Algebra: every non-constant polynomial has at least one complex root.
- Factor Theorem: if
ais a root of the polynomialf(x), then(x - a)is a factor off(x). - Remainder Theorem: if
f(x)is divided by(x - a), the remainder isf(a).
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Description
Learn about the process of expressing a polynomial as a product of simpler polynomials, including types of factorization over integers, rational numbers, and real numbers.
