Polynomial Division and Operations
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Questions and Answers

What is the term 3x² in the quotient the result of dividing 3x⁴ by?

What is the result of dividing x² + 3x + 1 by 3x² and bringing down 13x?

  • subtracting 3x⁴ + 9x³ + 3x² from the dividend and bringing down 13x
  • subtracting 3x⁴ + 9x³ + 3x² from the dividend and bringing down 13x. (correct)
  • multiplying 3x² by x² + 3x + 1
  • none of the above
  • Complete the division. The quotient is 3x² + x + ___

    -2 and 5

    If (3x² + 22x + 7) ÷ (x + 7) = 3x + 1, then (x + 7)(___) = ___.

    <p>3x + 1, 3x² + 22x + 7</p> Signup and view all the answers

    Identify the quotient and the remainder from (24x³ − 14x² + 20x + 6) ÷ (4x² − 3x + 5) = Q + R

    <p>Q = 6x + 1, R = -7x + 1</p> Signup and view all the answers

    What gives the width, w, of a rectangle when the area A is 120x² + 78x - 90 and the length l is 12x + 15?

    <p>10x - 6</p> Signup and view all the answers

    Divide. The quotient is ___ x + ___ and the remainder is ___ x² + ___ x + ___

    <p>10x, 16, 28, 10, 22</p> Signup and view all the answers

    Which expression represents the height of a rectangular prism given its volume 10x³ + 46x² - 21x - 27 and the area of the base 2x² + 8x - 9?

    <p>5x + 3</p> Signup and view all the answers

    How does checking polynomial division support the fact that polynomials are closed under multiplication and addition?

    <p>Polynomials are sums of terms with whole number exponents and are therefore closed under addition and multiplication.</p> Signup and view all the answers

    What is the remainder when (x³ - 2) is divided by (x - 1)?

    <p>-1</p> Signup and view all the answers

    Study Notes

    Polynomial Division Concepts

    • Long division of polynomials involves breaking down a polynomial, similar to numerical long division.
    • The term ( 3x^2 ) in the quotient arises when dividing ( 3x^4 ) by ( x^2 ).

    Polynomial Operations

    • The polynomial ( -2x^3 - x^2 + 13x ) is achieved after subtracting ( 3x^4 + 9x^3 + 3x^2 ) from a dividend and bringing down ( 13x ).
    • Completing the division gives a quotient of ( 3x^2 + x - 2 ) and a remainder of ( 5 ).

    Polynomial Multiplication Property

    • The equation ( (3x^2 + 22x + 7) ÷ (x + 7) = 3x + 1 ) exemplifies polynomial multiplication closure where ( (x + 7)(3x + 1) = 3x^2 + 22x + 7 ).

    Quotients and Remainders

    • For ( (24x^3 - 14x^2 + 20x + 6) ÷ (4x^2 - 3x + 5) ), the quotient is ( 6x + 1 ) and the remainder is ( -7x + 1 ).

    Area and Volume Applications

    • To find the width ( w ) of a rectangle with area ( A = 120x^2 + 78x - 90 ) and length ( l = 12x + 15 ), use division yielding ( w = 10x - 6 ).
    • The volume of a rectangular prism represented by ( 10x^3 + 46x^2 - 21x - 27 ) with a base area of ( 2x^2 + 8x - 9 ) results in a height ( h = 5x + 3 ).

    Closure Properties of Polynomials

    • Whole numbers are closed under addition; likewise, polynomials maintain closure under addition and multiplication since combining polynomials results in polynomials with whole number exponents.

    Remainder Theorem

    • Finding the remainder when dividing ( (x^3 - 2) ) by ( (x - 1) ) results in a remainder of ( -1 ).

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    Description

    This quiz covers key concepts in polynomial division, including long division, multiplication properties, and applications related to area and volume. It provides problem-solving practice for determining quotients and remainders of polynomial expressions.

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