All Questions ⁺⁰236111 Questions

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Polynomials

Let b be a constant. What is the smallest possible degree of the polynomial f(x) + b \cdot g(x), where f(x) = 2x^5 - 6x^4 - 4x^3 + 12x^2 + 7x - 5 and g(x) = x^6 - 17x^5 + 2x^4 + 6x^3 + 11x^2 - 8x + 1.

EquationEnthusiast ●

learnmgcat Jun 13, 2024

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21

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+1337

Polynomials

Let f be a cubic polynomial such that f(0) = 5, f(2) = 8, f(3)=13, and f(7) = -5. What is the sum of the coefficients of f?

learnmgcat Jun 13, 2024

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23

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+346

Geometry

In trapezoid ABCD, \overline{AB} \parallel \overline{CD}. Find the area of the trapezoid.

nathanl6656 Jun 13, 2024

Jun 12, 2024

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17

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+346

Geometry

Let $M$, $N$, and $P$ be the midpoints of sides $\overline{TU}$, $\overline{US}$, and $\overline{ST}$ of triangle $STU$, respectively. Let $\overline{UZ}$ be an altitude of the triangle. If $\angle TSU = 62^\circ$ and $\angle STU = 29^\circ$, read more ..

nathanl6656 Jun 12, 2024

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+346

Geometry

Altitudes $\overline{AD}$ and $\overline{BE}$ of acute triangle $ABC$ intersect at point $H$. If $\angle AHB = 144^\circ$ and $\angle BAH = 20^\circ$, then what is $\angle ABH$ in degrees?

nathanl6656 Jun 12, 2024

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20

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+346

Geometry

In triangle $STU$, let $M$ be the midpoint of $\overline{ST},$ and let $N$ be on $\overline{TU}$ such that $\overline{SN}$ is an altitude of triangle $STU$. If $ST = 13$, $SU = 14$, $TU = 4$, and $\overline{SN}$ and $\overline{UM}$ intersect at $X$, read more ..

nathanl6656 Jun 12, 2024

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+1337

Algebra

Determine all of the following for f(x) \cdot g(x), where f(x) = -7x^4-3x^2 + 2 and g(x) = 2x^8 - 3x^7 + 5x^6 + 3x^5 - 11x^4 + 3x^3 - 18x^2 + 17x - 5.
Leading term:
Leading coefficient:
Degree: read more ..

learnmgcat Jun 12, 2024

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14

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+1337

Algebra

There exists a polynomial f(x) and a constant k such that
(x^2 - 2x - 5) f(x) = x^4 - 2x^3 + kx^2
What is k?

learnmgcat Jun 12, 2024

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21

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+1337

Algebra

What is the coefficient of x in (x^3 + x^2 + x + 1)(x^4 + 3x^3 + 4x^2 + 8x + 7)?

NotThatSmart ●

learnmgcat Jun 12, 2024

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16

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+1337

Algebra

What is the coefficient of x^2 in (x^3 + x^2 + x + 1)(x^4 + 3x^3 + 8x + 7x + 1)?

NotThatSmart ●

learnmgcat Jun 12, 2024

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12

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+346

Algebra

Find the largest value of x for which
x^2 + y^2 = 4x + 5y
has a solution, if x and y are real.

NotThatSmart ●

nathanl6656 Jun 12, 2024

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11

1

+346

Algebra

Five workers have been hired to complete a job. If one additional worker is hired, they could complete the job $4$ days earlier. If the job needs to be completed $20$ days earlier, how many additional workers should be hired?

NotThatSmart ●

nathanl6656 Jun 12, 2024

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11

1

+346

Algebra

Three runners, Dirk, Edith, and Foley all start at the same time for a $24$ km race, and each of them runs at a constant speed. When Dirk finishes the race, Edith is $10$ km behind, and Foley is $15$ km behind. When Edith finishes the race, how far read more ..

nathanl6656 Jun 12, 2024

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16

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+346

Algebra

For a certain value of $k$, the system
x + y + 3z &= 10, \\
6x + 4y + z &= 18 - 5y - x, \\
kx + 7z &= 8 - 14y
has no solutions. What is this value of $k$?

nathanl6656 Jun 12, 2024

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15

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+346

Algebra

Find the ordered quintuplet (a,b,c,d,e) that satisfies the system of equations
a+2b+3c+4d+5e=47
2a+3b+4c+5d=15
3a+4b+5c=34 read more ..

nathanl6656 Jun 12, 2024

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14

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+1489

Number Theory

Find the smallest positive integer B so that when we express the decimal number 1000 as a base B number, we still get a 2-digit number.

kittykat Jun 12, 2024

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16

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+1557

Base arithmetic

Compute 12121_3 + 1222_3 in base 3.

parmen Jun 12, 2024

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16

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+1557

Base arithmetic

Compute 25_6 + 54_6 in base 6.

parmen Jun 12, 2024

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14

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+1557

Base arithmetic

Compute 35A811 - 275111 + A0411. Give your answer as a base 11 integer.
(You do not need to include the subscript to indicate the base.)

parmen Jun 12, 2024

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14

1

+1557

Base arithmetic

Compute 11_2 + 110_2 + 100111_2 + 101011_2 + 111_2 + 1011_2 + 1100101_2. Give your answer as a base 2 integer.

parmen Jun 12, 2024

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25

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+1557

Base arithmetic

Compute 2A3112 - B7512. Give your answer as a base 12 integer.

parmen Jun 12, 2024

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12

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+1557

Base arithmetic

Compute 3578 + 2238. Give your answer as a base 8 integer.

NotThatSmart ●

parmen Jun 12, 2024

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22

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+1337

Algebra

A mathematician works for t hours per day and solves p problems per hour, where t and p are positive integers. One day, the mathematician drinks some coffee and discovers that he can now solve 3p + 12 problems per hour. In fact, read more ..

learnmgcat Jun 12, 2024

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19

2

+1557

Counting

Franklin the fly starts at the point $(0,0)$ in the coordinate plane. At each point, Franklin takes a step to the right, left, up, or down. After $2$ steps, how many different points could Franklin end up at?

parmen Jun 12, 2024

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14

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+1557

Counting

Joanna has six beads that she wants to assemble into a bracelet. Four of the beads have the same color, and the other two all have different colors. In how many different ways can Joanna assemble her bracelet? (Two bracelets are considered identical read more ..

parmen Jun 12, 2024

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