Polynomial Long Division
Synthetic Division
Upper & Lower Bounds
Rational Zeros, Remainder, Factor Theorems
100
What is x^2-3x-10 divided by x+2
x-5
100
What is 2x^2-x-3 divided by x+1
2x-3
100
Use synthetic division to prove that the number 3 is an upper bound for the real zeros of the function f(x)=2x^3-4x^2+x-2
The numbers in the last line which are 2, 2, 7, and 19 are all nonnegative so 3 is an upper bound.
100
Use the factor theorem to determine whether x-1 is a factor of x^3-x^2+x-1
Yes, x-1 is a factor.
200
What is 2x^2-5x-1 divided by x-3
2x+1
200
What is x^4-10000 divided by x-10
x^3+10x^2+100x+1000
200
Use synthetic division to prove that the number 2 is an upper bound for the real zeros of the function x^4-x^3+x^2+x-12
The numbers in the last line which are 1, 1, 3, 7, 2 are all nonnegative so 2 is an upperbound.
200
Use the factor theorem to determine whether x-2 is a factor of x^3+3x-4
No, x-2 is not a factor.
300
What is x^6+2x^4+6x-9 divided by x^3+3
x^3+2x-3
300
What is 2x^4-1x^3-7x^2-3x+10 divided by x-2
2x^3+3x^2-1x-5
300
Use synthetic division to prove that the number -1 is a lower bound for the real zeros of the function 3x^3-4x^2+x+3
The numbers in the last line which are 3, -7, 8, -5 have alternating signs so -1 is a lower bound.
300
Use the remainder theorem to find the remainder when 2x^2-3x+1 is divided by 2
The remainder is 3
400
What is 4x^3+2x^2-10x divided by 2x^2-4
2x+1+ -2x+4/2x^2-4
400
What is 2x^4-9x^3+21x^2-26x-12 divided by 2x-3
x^3-3x^2+6x-4
400
Use synthetic division to prove that the number 0 is a lower bound for the real zeros of the function x^3-4x^2+7x-2
The numbers in the last line which are 1, -4, 7, -2 have alternating signs so 0 is a lower bound.
400
Use the remainder theorem to find the remainder when x^3-x^2+2x-1 is divided by -3
The remainder is -43
500
What is x^2+2x^2y-2xy+2xy^2-3y divided by x+y
x+2xy-3y
500
What is x^6+x^4+0x^4+0x^3-x^2+0x+2 divided by x^3+0x^2+x+2
x^3+x^2-x-3 R:-2x^3+5x+8
500
Use synthetic division to prove that the number 5 is an upper bound for the real zeros of the function 2x^3-5x^2-5x-1
The numbers in the last line which are 2, 5, 20, 99 are all nonnegative so 5 is an upper bound.
500
Use the rational zeros theorem to write a list of all potential zeros. Then determine which ones, if any, are zeros of the function f(x)=6x^3-5x-1
1, 1/2, 1/3, 1/6, -1, -1/2, -1/3, -1/6 1 is a zero.
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