All Questions ⁺⁰237779 Questions

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25

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+1022

Algebra

Factor x^2 - 2x - y^2 + 2yz + 5z^2 as the product of two polynomials of degree 1.

gnistory Mar 2, 2025

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25

0

+456

Algebra

Suppose the real numbers $a$, $b$, $x$, and $y$ satisfy the equations
ax + by = 3,
ax^2 + by^2 = 5,
ax^3 + by^3 = 17,
ax^4 + by^4 = 23.<#> Evaluate ax^5 + by^5.

Jasmo Mar 2, 2025

0

23

0

+456

Algebra

Let a, b, c, and d be distinct real numbers such that
a = \sqrt{4 + \sqrt{5 + a}},
b = \sqrt{4 - \sqrt{7 + b}},
c = \sqrt{4 + \sqrt{9 - c}},
d = \sqrt{4 - \sqrt{11 - d}}.
Compute abcd.

Jasmo Mar 2, 2025

Mar 1, 2025

0

25

0

+456

Algebra

If a_1, a_2, ..., a_19 satisfy
a_1 + a_2 + a_3 + a_4 = 1,
a_2 + a_3 + a_4 + a_5 = 2,
a_3 + a_4 + a_5 + a_6 = 3,
...
a_{16} + a_{17} + a_{18} + a_{19} = 16,
a_{17} + a_{18} + a_{19} + read more ..

Jasmo Mar 1, 2025

0

25

0

+456

Algebra

Let a = 3(x - y), b = 3(y - z), and c = 3(z - x), where x, y, z are real numbers, and assume ab + ac + bc \ne 0. Compute
(a^3 + b^3 + c^3)/(ab + ac + bc).

Jasmo Mar 1, 2025

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1

14

1

+900

Algebra

Let r_1, r_2, r_3, r_4, and r_5 be the complex roots of x^5 - 4x^2 + 7x - 1 = 0. Compute
(r_1^2 + r_1^6 + 2)(r_2^2 + r_2^6 + 2)(r_3^2 + r_3^6 + 2)(r_4^2 + r_4^6 + 2)(r_5^2 + r_5^6 + 2)

eramsby1O1O Mar 1, 2025

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22

0

+900

Algebra

Fill in the blanks, to complete the factorizaion:
(a^2 + b^2 - c^2)^2 - 4a^2 b^2 - 4a^2 c^2 + 4b^2 c^2 = (a + ___)(a + ___)(a + ___)(a + ___)

eramsby1O1O Mar 1, 2025

0

21

1

+900

Algebra

Fill in the blank with a constant, so that the resulting expression can be factored as the product of two linear expressions:
2mn - 18m + 5n - mn + 20m + 4n + ___

eramsby1O1O Mar 1, 2025

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26

0

+900

Algebra

Let t be a root of f(x) = x^3 - x + 3. Evaluate t^6 - 4t^5 + 7t^4 - 3t^2 + 10t - 13.

eramsby1O1O Mar 1, 2025

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26

0

+519

Algebra

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)

bIueb3rry Mar 1, 2025

0

15

0

+519

Algebra

The real numbers x and y satisfy
x^2 + y^2 - 8x + 6y + 23 = 0.
Find the largest possible value of x + y.

bIueb3rry Mar 1, 2025

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12

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+519

Algebra

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

bIueb3rry Mar 1, 2025

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25

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+519

Algebra

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

bIueb3rry Mar 1, 2025

0

31

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+381

Algebra

Let a, b, c be the roots of p(x) = x^3 + 7x^2 + 10x - 13 - 5x^3 + 25x^2 + 44x - 15. Find the value of
1/(ab + ac) + 1/(ab + bc) + 1/(ac + bc)

cwimie Mar 1, 2025

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1

19

1

+381

Algebra

Suppose the polynomial p(x)=x^3+ax^2+bc+c has the property that the mean of its zeroes, the product of its zeroes, and the sum of its coefficients are all equal. If the y-intercept of the graph of y=p(x) is 0, what is b?

cwimie Mar 1, 2025

0

12

1

+381

Algebra

Find all (real or nonreal) x satisfying
(x - 3)^4 + (x - 5)^4 = -8 + 6(x - 3)(x - 5)^3 - 11(x - 3)^3 (x - 5).

cwimie Mar 1, 2025

Feb 28, 2025

Feb 27, 2025

0

29

0

+456

Algebra

Express x^8 + x^4 y^4 - x^6 y^2 + x^3 y^5 - 4xy^7 - y^8 as the product of two polynomials of degree 2, and a polynomial of degree 4.

Jasmo Feb 27, 2025

0

19

1

+456

Algebra

Find the constant k so that
2x^2 + 5xy - 8y^2 + 7x + 25y + k
can be expressed as the product of two linear factors of the form ax + by + c.

Jasmo Feb 27, 2025

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2

23

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+381

Algebra

Let
p(a, b, c) = a^5 + b^5 + c^5 + k(a^3 + b^3 + c^3)(a^4 + b^4 + c^4 - 2a^2 b^2 - 2a^2 c^2 - 2b^2 c^2).
Find the constant k so that p(a,b,c) is divisible by a + b + c.

cwimie Feb 27, 2025

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1

8

1

+381

Algebra

If z is a complex number satisfying
z + \frac{1}{z} = \sqrt{2},
calculate
z^{10} + \frac{1}{z^{10}}. read more ..

cwimie Feb 27, 2025

Feb 26, 2025

0

2

14

1

+381

Algebra

In this multi-part problem, we will consider this system of simultaneous equations:
3x + 4y + 30z = -60,
2xy + 42xz - 16yz = 68,
5xyz = 56.
Let a = x/2, b = 5y and c = -4z.
Determine the monic cubic polynomial in terms of read more ..

cwimie Feb 26, 2025

0

2

24

1

+381

Algebra

In this multi-part problem, we will consider this system of simultaneous equations:
3x + 4y + 30z = -60,
2xy + 42xz - 16yz = 68,
5xyz = 56.
Let a = x/2, b = 5y and c = -4z.
Determine the monic cubic polynomial in terms of read more ..

cwimie Feb 26, 2025

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