Polynomials

What is Algebra?

Algebra is the branch of mathematics that uses letters instead of unknown numbers.

These letters are called variables or unknowns.

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What is an algebraic expression?

An Algebraic Expression is a combination of letters (variables) and numbers using the operations of addition, subtraction, multiplication, division and exponentiation.

An algebraic expression is made up of terms.

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Examples:

Monomial: It can be a number, a variable or the product of a number and one or more variables. A monomial has two parts:

• Coefficient: The number usually written in front of the variable(s), that multiplies it (them).
• Literal part: All the variables in a monomial with their corresponding exponents.

Degree of a monomial: It is the sum of the exponents of all of the variables in the monomial.

Vocabulary

Examples:

Coefficient: 7

Literal part: No one

Degree: 0

No coefficient, its value is 1

Coefficient: 2

Literal part:

Degree: 7

Coefficient: 9

Literal part:

Degree: 3

Coefficient: -2

Literal part:

Degree: 1

Coefficient:

Literal part:

Degree: 3

Coefficient: 1

Literal part:

Degree: 5

Vocabulary

Polynomial: It can be a monomial or a sum of monomials. They can be classified by its number of terms:

• Monomial: A polynomial with only one term.
• Binomial: A polynomial with exactly two terms.
• Trinomial: A polynomial with exactly three terms.
• Polynomial: It is usually the name given to a polynomial that has more than 4 or equal to 4 degree.

Vocabulary

Classify polynomials by

number or terms:

Degree of a polynomial : It is the largest exponent of the terms that forms the polynomial (the biggest degree of the monomials in the polynomial) . A polynomial can be classified by its degree:

• Constant: A polynomial of degree “zero”.
• Linear: A polynomial of 1^st degree.
• Quadratic: A polynomial of 2^nd degree.
• Cubic: A polynomial of 3^rd degree.
• Quartic: A polynomial of 4^th degree.
• Quintic: A polynomial of 5^th degree, and so on.

Vocabulary

Classify polynomials by

degree:

Vocabulary

• Polynomial in one variable: It is a polynomial with only one variable in all its terms.
• Standard form: A polynomial is written in standard form when its terms are arranged from the largest degree term to the smallest degree term (that is, in descending order regarding the degrees).
• Leading term: It is the first term on a polynomial in standard form. (It is the monomial that has the biggest degree).
• Leading coefficient: It is the coefficient of the leading term.
• Constant (or independent term): The coefficient of zero degree. (The “number” without any variable)

Write the polynomials in standard form.

Examples:

Leading term

Leading coefficient: 4

Polynomial degree: 4

Leading term

Leading coefficient: -1

Polynomil degree: 5

Vocabulary

• Complete polynomial: They are polynomials with non-negative coefficient for each exponent of the variable starting with the degree of the leading term.
• Symplified polynomial or a polynomial in simplest form: It is a polynomial that does NOT have any like monomials.

Incomplete. There is NO 1st degree term.

Complete polynomial

Rewrite the polynomials in standard form. Calculate its leading term, its leading coefficient and its degree.

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Standard form:

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Examples:

Leading

term

Leading

coefficient

Polynomial degree

Constant term

Rewrite the polynomials in standard form. Calculate its leading term, its leading coefficient and its degree.

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Standard form:

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Examples:

Leading

term

Leading

coefficient

Polynomial degree

Constant term

Rewrite the polynomials in standard form. Calculate its leading term, its leading coefficient and its degree.

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Standard form:

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Examples:

Leading

term

Leading

coefficient

Polynomial degree

Constant term