All Questions ⁺⁰237784 Questions

Sep 10, 2025

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Math

Suppose f(x) is a rational function such that
3 f \left( \frac{1}{x} \right) - \frac{f(x)}{x} = x
for all $x \neq 0$. Find f(-2).

learnmgcat Sep 10, 2025, 3:50:18 PM

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3

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+1540

Math

Let F(x) be the real-valued function defined for all real x except for x = 1 and x = 2 and satisfying the functional equation
F(x) + F \left( \frac{2x - 3}{x - 1} \right) + F \left( \frac{1}{x} \right) = x.
Find the function F(x) satisfying read more ..

learnmgcat Sep 10, 2025, 3:50:00 PM

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+1540

Math

The function f(n) is defined for all integers n, such that
f(x) + f(y) = f(x + y) - 4xy - 1 + f(x^2) + f(y^2)
for all integers x and y, and f(1) = 1. Find f(n).

learnmgcat Sep 10, 2025, 3:49:40 PM

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+1540

Math

The function f(n) takes the integers to the real numbers such that
f(m + n) + f(m - n) = 2f(m) + 2f(n) + mn
for all integers m and n, and f(1) = 2. Find f(n).

learnmgcat Sep 10, 2025, 3:49:26 PM

Sep 8, 2025

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4

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+1022

Math

The polynomial
g(x) = x^3 - 14x^2 - 67x - 90 + 5x^3 - 20x^2 + 7x - 135
has one integer root. What is it?

gnistory Sep 8, 2025

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5

1

1

+1022

Math

If f is a polynomial of degree 4 such that
f(0) = 1, f(1) = 2, f(2) = -7, f(3) = 0, f(4) = 3,
then determine f(5).

gnistory Sep 8, 2025

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5

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2

+1022

Math

Let f(x) be a polynomial with integer coefficients. There exist distinct integers p, q, r, s, t such that
f(p) = f(q) = f(r) = f(s) = 18
and f(t) > 18. What is the smallest possible value of f(t)?

gnistory Sep 8, 2025

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5

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1

+1022

Math

Find all positive integers n for which n^2 - 19n + 59 + n^2 + 4n - 31 is a perfect square.

gnistory Sep 8, 2025

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5

1

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+519

Math

Let f(x) = x^3 + ax^2 + bx + c be a polynomial. All the roots are negative integers. If a + b + c = 10, then find c.

bIueb3rry Sep 8, 2025

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5

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+519

Math

Let f(x) = x^3 + ax^2 + bx + c be a polynomial. All the roots of f(x) are negative integers. If a + b + c = 10, then find the roots of f(x).

bIueb3rry Sep 8, 2025

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5

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+519

Math

Let p(x) be a polynomial with integer coefficients. Suppose p(4) = 5 and p(-2) = 3. If p(x) has an integer root r, then find all possible values of r.

bIueb3rry Sep 8, 2025

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4

1

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+519

Math

Let p(x) be a cubic polynomial. If p(0) = 0, p(1) = 1, p(2) = 2, and p(3) = 3, then compute p(4).

bIueb3rry Sep 8, 2025

Sep 7, 2025

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6

1

0

+1013

Math

Find all roots of the polynomial
f(x) = x^4 - 5x^3 + 5x^2 + 17x - 42 + 4x^4 + 10x^3 - 18x^2 + 2x - 5.

magenta Sep 7, 2025

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7

1

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+1013

Math

Find a monic quartic polynomial f(x) with rational coefficients whose roots include x = 1 - \sqrt{2} and x = 2 and x = 7. Give your answer in expanded form.

magenta Sep 7, 2025

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7

1

1

+1013

Math

Let f(x) = x^3 + bx + c, where b and c are integers. If f(\sqrt{5}) = 0, determine b + c.

magenta Sep 7, 2025

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7

1

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+1013

Math

Find a monic quartic polynomial f(x) with rational coefficients whose roots include x = 2 - i \sqrt[3]{3}$. Give your answer in expanded form.

magenta Sep 7, 2025

Sep 6, 2025

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9

1

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+381

Math

Find the conic section represented by the equation
5x^2 + y^2 + 10x - 4y + 17 = -4y^2 - 18x + 12y

cwimie Sep 6, 2025

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8

1

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+381

Math

Find the conic section represented by the equation
4x^2 - 8y^2 - 28x + 49 = 4x^2 + 5x - 11y + 20

cwimie Sep 6, 2025

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9

1

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+381

Math

Find the conic section represented by the equation
-x^2 + 2y^2 - 8x + 10y - 43 = 3y^2 - 17x + 14

cwimie Sep 6, 2025

0

8

1

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+381

Math

Find the foci of the ellipse shown below.
Enter your answer as a list of ordered pairs separated by commas.
Points on the ellipse are (-4,9), (-7,2), (-5,2), (-7, 1)

cwimie Sep 6, 2025

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9

1

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+16

Math

Parts (a)-(c) refer to the graph of
-4x^2 + y^2 - 2y = -3y^2 + 8x + 9y + 15
which is a hyperbola.
Find the center of the hyperbola.

keinaku Sep 6, 2025

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9

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+16

Math

Find the vertices of the hyperbola from part (a).
-4x^2 + y^2 - 2y = -3y^2 + 8x + 9y + 15

keinaku Sep 6, 2025

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9

1

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+16

Math

Find the asymptote of the hyperbola from part (a) with positive slope.
Enter your answer as an equation of the form $y = mx + b$.
-4x^2 + y^2 - 2y = -3y^2 + 8x + 9y + 15

keinaku Sep 6, 2025

0

8

1

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+16

Math

Find the constant k so that the equation
4x^2 + 9y^2 - 8x + 54y + k = 2x^2 + 5y^2 - 12x + 34y
represents an ellipse which has an area of 6 \pi.

keinaku Sep 6, 2025

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8

1

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+519

Math

Find the quotient and remainder when p(x) is divided by q(x), where p(x) = x^2 + 2 and q(x) = x^4 + 11x^2 - 7x + 10.

bIueb3rry Sep 6, 2025

0

9

1

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+519

Math

Find the quotient and remainder when p(x) is divided by q(x)$, where p(x) = x^4 - x^2 + 3x - 7 and q(x) = x^2 + 2x - 5.

bIueb3rry Sep 6, 2025

Sep 5, 2025

0

9

1

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+1309

Math

Find an ordered pair of constants (a,b) such that the polynomial f(x)=x^3+ax^2+(b+2)x+1 is divisible by x^2-3.
Enter your answer as an ordered pair in the format $(a,b)$, where $a$ and $b$ are replaced by appropriate numbers.

AnswerscorrectIy Sep 5, 2025

0

9

3

0

+1309

Math

Given that x = 2 is a root of p(x) = x^4 - 3x^3 + 11x^2 - 25x + k for some constant k, find all the roots of p(x) (real and nonreal).
Enter your answer as a list of numbers separated by commas.

AnswerscorrectIy Sep 5, 2025

0

10

1

0

+1309

Math

Find the remainder when p(x) is divided by q(x), where p(x) = x^5 + 1 and q(x) = x^2 + x + 2.

AnswerscorrectIy Sep 5, 2025

0

10

1

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+1580

Math

The polynomial
g(x) = x^3 - x^2 - (m^2 + m + 18) x + 2m^2 - 14m - 6
is divisible by x - 5 and all of its zeroes are integers. Find all possible values of m.

kittykat Sep 5, 2025

0

11

1

0

+1580

Math

Let f(x) be a polynomial such that f(0) = 4, f(1) = 6, and f(2) = -2. Find the remainder when f(x) is divided by x.

kittykat Sep 5, 2025

0

9

1

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+1580

Math

Let f(x) be a polynomial such that f(0) = 4, f(1) = 8, and f(2) = -6. Find the remainder when f(x) is divided by x.

kittykat Sep 5, 2025

0

11

1

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+1580

Math

Let f(x) = x^{10}+5x^9-8x^8+4x^7-x^6-33x^5+4x^4-22x^3+8x^2-3x-5.
Without using long division (which would be horribly nasty!), find the remainder when f(x) is divided by x^2 - 4.

kittykat Sep 5, 2025

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