9 Questions
What is the process of expressing a polynomial as a product of simpler polynomials called?
Factorization
Which type of factorization involves factoring a polynomial into factors with rational coefficients?
Factorization over rational numbers
What is the first step in the method of factoring out the greatest common factor (GCF) of a polynomial?
Find the GCF of all terms in the polynomial
What is the formula used in factoring quadratic polynomials?
x^2 + bx + c = (x + d)(x + e)
What is the result stated by the Fundamental Theorem of Algebra?
Every non-constant polynomial has at least one complex root
What is the statement of the Factor Theorem?
If a is a root of the polynomial f(x), then (x - a) is a factor of f(x)
What method of factorization involves expressing the polynomial as a sum of simpler polynomials and then factoring each simpler polynomial?
Factoring by decomposition
What is the first step in the method of factoring by grouping?
Group terms with common factors
What is the purpose of factorization of polynomials?
To express the polynomial as a product of simpler polynomials
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
d
ande
that 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
a
is 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
d
ande
that 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
a
is 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)
.
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.
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