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Questions and Answers
What is the term 3x² in the quotient the result of dividing 3x⁴ by?
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?
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 + ___
Complete the division. The quotient is 3x² + x + ___
-2 and 5
If (3x² + 22x + 7) ÷ (x + 7) = 3x + 1, then (x + 7)(___) = ___.
If (3x² + 22x + 7) ÷ (x + 7) = 3x + 1, then (x + 7)(___) = ___.
Identify the quotient and the remainder from (24x³ − 14x² + 20x + 6) ÷ (4x² − 3x + 5) = Q + R
Identify the quotient and the remainder from (24x³ − 14x² + 20x + 6) ÷ (4x² − 3x + 5) = Q + R
What gives the width, w, of a rectangle when the area A is 120x² + 78x - 90 and the length l is 12x + 15?
What gives the width, w, of a rectangle when the area A is 120x² + 78x - 90 and the length l is 12x + 15?
Divide. The quotient is ___ x + ___ and the remainder is ___ x² + ___ x + ___
Divide. The quotient is ___ x + ___ and the remainder is ___ x² + ___ x + ___
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?
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?
How does checking polynomial division support the fact that polynomials are closed under multiplication and addition?
How does checking polynomial division support the fact that polynomials are closed under multiplication and addition?
What is the remainder when (x³ - 2) is divided by (x - 1)?
What is the remainder when (x³ - 2) is divided by (x - 1)?
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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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