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Polynomials: Operations and Properties
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Polynomials: Operations and Properties

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Questions and Answers

If f(x) = x^3 - 2x^2 - 5x + 6 and f(x) is divided by (x - 1), what is the remainder?

  • 4
  • 2 (correct)
  • 3
  • 5
  • Which of the following is a factorization of x^2 + 5x + 6?

  • (x + 1)(x - 6)
  • (x + 1)(x + 6)
  • (x - 2)(x - 3)
  • (x + 2)(x + 3) (correct)
  • What is the result of (x^2 + 2x - 1) - (x^2 - 3x - 2)?

  • 5x - 1
  • 5x + 1
  • 5x - 3 (correct)
  • 5x + 3
  • If (x + 2)(x - 3) = x^2 - x - 6, what is the product of (x + 2) and (x - 3)?

    <p>x^2 - x - 6</p> Signup and view all the answers

    What is the result of (2x^2 + 3x - 1) + (x^2 - 2x - 3)?

    <p>3x^2 + x - 4</p> Signup and view all the answers

    If f(x) = x^2 + 2x - 3 and f(a) = 5, what is the value of a?

    <p>2</p> Signup and view all the answers

    Study Notes

    Polynomials

    Addition of Polynomials

    • To add two or more polynomials, combine like terms
    • Add corresponding coefficients
    • Example: (2x^2 + 3x - 1) + (x^2 - 2x - 3) = 3x^2 + x - 4

    Subtraction of Polynomials

    • To subtract one polynomial from another, change the sign of the polynomial being subtracted and add
    • Example: (2x^2 + 3x - 1) - (x^2 - 2x - 3) = x^2 + 5x + 2

    Multiplication of Polynomials

    • To multiply two polynomials, multiply each term of one polynomial by each term of the other
    • Combine like terms
    • Example: (2x + 3)(x + 2) = 2x^2 + 7x + 6

    Factorization of Polynomials

    • Factorization: expressing a polynomial as a product of simpler expressions
    • Methods:
      • Factoring out greatest common factor (GCF)
      • Factoring quadratic expressions (e.g., difference of squares)
      • Factoring sum or difference of cubes
    • Example: x^2 + 5x + 6 = (x + 3)(x + 2)

    Remainder Theorem

    • If a polynomial f(x) is divided by (x - a), the remainder is f(a)
    • Example: if f(x) = x^2 + 2x - 3, then f(2) = 2^2 + 2(2) - 3 = 5, which is the remainder when dividing f(x) by (x - 2)
    • The Remainder Theorem is useful for evaluating polynomials and finding roots.

    Polynomials

    Addition of Polynomials

    • Combine like terms to add two or more polynomials
    • Add corresponding coefficients to get the resulting polynomial

    Subtraction of Polynomials

    • Change the sign of the polynomial being subtracted and add to get the resulting polynomial
    • Subtracting a polynomial is equivalent to adding its negative

    Multiplication of Polynomials

    • Multiply each term of one polynomial by each term of the other to get the product
    • Combine like terms to simplify the resulting polynomial

    Factorization of Polynomials

    • Factorization is expressing a polynomial as a product of simpler expressions
    • Methods of factorization include:
      • Factoring out the greatest common factor (GCF)
      • Factoring quadratic expressions (e.g., difference of squares)
      • Factoring sum or difference of cubes
    • Factorization can be used to simplify polynomials and solve equations

    Remainder Theorem

    • The remainder of dividing a polynomial f(x) by (x - a) is f(a)
    • The Remainder Theorem is useful for evaluating polynomials and finding roots
    • The theorem can be used to find the remainder of dividing a polynomial by a linear factor

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    Description

    This quiz covers the basics of polynomial operations, including addition, subtraction, and multiplication of polynomials.

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