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?

    <p>The greatest degree of any term in the polynomial.</p> Signup and view all the answers

    How do you find the degree of a monomial?

    <p>The sum of the exponents of all the variables in a monomial.</p> Signup and view all the answers

    What is the standard form of a polynomial?

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

    What are leading coefficients?

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

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

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

    What is a quadratic expression?

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

    What does factoring mean?

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

    What is the zero product property?

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

    What is a quadratic equation?

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

    What is a prime polynomial?

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

    What is the difference of two squares?

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

    What is a perfect square trinomial?

    <p>A trinomial that is the square of a binomial.</p> Signup and view all the answers

    How can a quadratic equation be solved?

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

    How is the degree of a polynomial determined?

    <p>The highest exponent is the same as your degree.</p> Signup and view all the answers

    What is a constant in terms of polynomial degree?

    <p>It is known as a 0 degree polynomial.</p> Signup and view all the answers

    What characterizes a first degree polynomial?

    <p>It is known as a linear polynomial.</p> Signup and view all the answers

    Define a second degree polynomial.

    <p>It is known as a quadratic polynomial.</p> Signup and view all the answers

    What is a third degree polynomial called?

    <p>It is known as a cubic polynomial.</p> Signup and view all the answers

    Describe a fourth degree polynomial.

    <p>It is known as a quartic polynomial.</p> Signup and view all the answers

    What is a quintic polynomial?

    <p>It is known as a fifth degree polynomial.</p> Signup and view all the answers

    How does one identify terms in a polynomial?

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

    How do you name a polynomial?

    <p>Combine the degree and more specific group of polynomials.</p> Signup and view all the answers

    What is a leading term in a polynomial?

    <p>The term that has the highest exponent.</p> Signup and view all the answers

    What are coefficients in a polynomial?

    <p>The number in front of the variable is the coefficient.</p> 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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