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Questions and Answers
What is the quotient when dividing 2x^3 + x^2 + x by x?
In the polynomial division of 3x^2 + x + 1 by x, what is the remainder?
What are the factors of 2x^3 + x^2 + x?
Is x a factor of 3x^2 + x + 1?
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What is the quotient when dividing p(x) = x + 3x^2 - 1 by g(x) = 1 + x?
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When dividing a polynomial by a monomial, what can be said about the terms and the divisor?
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When dividing 3x^2 + x + 1 by x, why do we stop at 1 as the remainder?
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In the polynomial division example provided, what is considered as the quotient and what as the remainder?
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Why can't x be considered a factor of 2x^3 + x^2 + x?
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"In fact, you may have noticed that 'x' is common to each term of 2x^3 + x^2 + x." This statement implies what about the terms of the polynomial?
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Study Notes
Polynomial Division Process
- Start by dividing the first term of the dividend by the first term of the divisor to obtain the first term of the quotient.
- For example, ( \frac{3x^2}{x} = 3x ).
- Multiply the entire divisor by the first term of the quotient, then subtract this product from the original dividend.
Calculating the Remainder
- Multiplying the divisor ( (x + 1) ) by ( 3x ) results in ( 3x^2 + 3x ).
- Subtracting this from the dividend ( 3x^2 + x - 1 ) yields a remainder of ( -2x - 1 ).
Continued Division
- Treat the remainder ( -2x - 1 ) as the new dividend for the next division step.
- Divide the first term of the new dividend ( -2x ) by the first term of the divisor ( x ) to obtain the second term of the quotient, ( -2 ).
Further Calculation Steps
- Multiply the divisor ( (x + 1) ) by ( -2 ) resulting in ( -2x - 2 ).
- Subtract this from the new dividend ( -2x - 1 ) to get the remainder 1.
Completion of the Division
- The division process continues until the remainder is either 0 or the degree of the new dividend is less than the degree of the divisor.
- The sum of all terms obtained during the process becomes the complete quotient.
Final Result
- The complete quotient from the example is ( 3x - 2 ) with a remainder of 1.
- The equation ( 3x^2 + x - 1 = (x + 1)(3x - 2) + 1 ) illustrates the relationship between dividend, divisor, quotient, and remainder.
General Polynomial Division Rule
- If ( p(x) ) and ( g(x) ) are two polynomials where the degree of ( p(x) ) is greater than or equal to the degree of ( g(x) ), we can express the division as:
- ( p(x) = g(x)q(x) + r(x) )
- where ( r(x) ) is either 0 or has a degree less than that of ( g(x) ).
Relation Between Remainder and Polynomial Value
- If ( p(x) = 3x^2 + x - 1 ), substituting ( x = -1 ) gives ( p(-1) = 1 ).
- The result demonstrates that the remainder of the polynomial division matches the value of ( p(x) ) evaluated at the root of the divisor ( x + 1 ).
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Description
Learn the step-by-step process of dividing polynomials using the long division method. Understand how to find the quotient and remainder by dividing the terms of the dividend by the terms of the divisor.
