Algebra Chapter 8: Quadratic Expressions

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Questions and Answers

What is a polynomial?

A binomial

Both A and B (correct)

The sum of monomials

A monomial

What is a binomial?

The sum of two monomials and a type of polynomial.

What is a trinomial?

The sum of three monomials and a type of polynomial.

What does the degree of a polynomial refer to?

The greatest degree of any term in the polynomial. Signup and view all the answers

How do you find the degree of a monomial?

The sum of the exponents of all the variables in a monomial. Signup and view all the answers

What is the standard form of a polynomial?

A polynomial that is written with the terms in order from greatest degree to least degree. Signup and view all the answers

What are leading coefficients?

The coefficient of the term with the highest degree in a polynomial. Signup and view all the answers

What is the F.O.I.L method?

To multiply two binomials, find the sum of the products of the First terms, the Outer terms, the Inner terms, and the Last terms. Signup and view all the answers

What is a quadratic expression?

An expression in one variable with a degree of 2 written in the form of ax^2 + bx + c. Signup and view all the answers

What does factoring mean?

To express a polynomial as the product of monomials and polynomials. Signup and view all the answers

What is the zero product property?

If the product of two factors is zero, then at least one of the factors must be zero. Signup and view all the answers

What is a quadratic equation?

An equation that can be written in the form ax^2 + bx + c = 0, where a is not zero. Signup and view all the answers

What is a prime polynomial?

A polynomial that cannot be written as a product of two polynomials with integral coefficients. Signup and view all the answers

What is the difference of two squares?

Two perfect squares that are separated by a subtraction sign like this: a^2 - b^2. Signup and view all the answers

What is a perfect square trinomial?

A trinomial that is the square of a binomial. Signup and view all the answers

How can a quadratic equation be solved?

By taking the square root of both sides of an equation. Signup and view all the answers

How is the degree of a polynomial determined?

The highest exponent is the same as your degree. Signup and view all the answers

What is a constant in terms of polynomial degree?

It is known as a 0 degree polynomial. Signup and view all the answers

What characterizes a first degree polynomial?

It is known as a linear polynomial. Signup and view all the answers

Define a second degree polynomial.

It is known as a quadratic polynomial. Signup and view all the answers

What is a third degree polynomial called?

It is known as a cubic polynomial. Signup and view all the answers

Describe a fourth degree polynomial.

It is known as a quartic polynomial. Signup and view all the answers

What is a quintic polynomial?

It is known as a fifth degree polynomial. Signup and view all the answers

How does one identify terms in a polynomial?

Terms are always separated by addition or subtraction, NOT multiplication or division. Signup and view all the answers

How do you name a polynomial?

Combine the degree and more specific group of polynomials. Signup and view all the answers

What is a leading term in a polynomial?

The term that has the highest exponent. Signup and view all the answers

What are coefficients in a polynomial?

The number in front of the variable is the coefficient. Signup and view all the answers

Study Notes

Polynomial Concepts

A polynomial is defined as a monomial or a sum of monomials.
Binomial refers to the sum of two monomials, while a trinomial includes three monomials. Both are specific types of polynomials.
The degree of a polynomial is determined by the highest degree of its terms.
The degree of a monomial is the sum of all exponents of its variables.

Polynomial Form and Structure

Standard form arranges polynomial terms from the highest degree to the lowest degree.
The leading coefficient is the coefficient of the term with the highest degree in a polynomial.

Operations with Polynomials

The F.O.I.L. method is used for multiplying two binomials, focusing on the products of their First, Outer, Inner, and Last terms.
Factoring involves expressing a polynomial as a product of monomials or other polynomials.
Factoring by grouping applies the Distributive Property to factor polynomials with four or more terms.

Quadratics and Equations

A quadratic expression is a polynomial of degree 2, typically presented as (ax^2 + bx + c).
A quadratic equation is an equation in the form (ax^2 + bx + c = 0), with (a \neq 0).
The zero product property states that if the product of two factors equals zero, at least one factor must be zero.

Special Polynomial Forms

A prime polynomial cannot be factored into two polynomials with integer coefficients.
The difference of two squares is represented as (a^2 - b^2).
A perfect square trinomial arises from squaring a binomial.

Solving Quadratic Equations

The square root property allows solving quadratic equations by taking square roots of both sides, requiring both positive and negative solutions.

Degrees of Polynomials

0 degree polynomials are constants (e.g., 3).
1st degree polynomials are linear (e.g., (2x + 1)).
2nd degree polynomials are quadratic (e.g., (x^2 - 4x)).
3rd degree polynomials are cubic (e.g., (y^3 + y^2 - 4)).
4th degree polynomials are quartic (e.g., (x^4 - 3)).
5th degree polynomials are quintic (e.g., (y^5 + y^2)).
Higher degree polynomials are classified by their respective degree (e.g., (z^6 - z^3) for 6th degree).

Identifying Polynomial Terms

Terms within a polynomial are separated by addition or subtraction.
Naming polynomials involves combining their degree with the type (e.g., a cubic binomial).

Key Components of Polynomials

The leading term has the highest exponent in a polynomial.
Coefficients are the numbers in front of variables; if no number is present, it is inferred to be 1. For example, in (-5x + 3x^2 - 12 + x^3), coefficients are (-5), (3), and (1) respectively.
The leading coefficient is the coefficient of the leading term.

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Test your knowledge of key terms related to quadratic expressions and equations in this flashcard quiz. Each card includes a specific mathematical term alongside its definition, helping reinforce your understanding of polynomials, binomials, and more.