Polynomial Division and Operations

Questions and Answers

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

x²

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)(_) = _.

3x + 1, 3x² + 22x + 7 Signup and view all the answers

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

Q = 6x + 1, R = -7x + 1 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?

10x - 6 Signup and view all the answers

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

10x, 16, 28, 10, 22 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?

5x + 3 Signup and view all the answers

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

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

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

-1 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 ).

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.