a^3 – 3a^2b + 3ab^2 – b^3
________________________
(a – b )(a – b)
If I did not recognize that the numerator was the difference of cubes and that I could factor out (a-b)^2 from numerator and denominator, could this be solved using polynomial division?
Here’s a question from the May 2023 (US) SAT, non-calculator section:
Alice took 60 minutes to complete a task on her first trial. The time it took Alice to complete the task decreased by 10% of the previous time for each additional trial. Approximately how many minutes will it take Alice to complete the task on her fifth trial?
A) 50
B) 42
C) 39
D) 35
I know we can solve this by subtracting 10% of the current value 5 times, but is there a more direct way to solve, maybe as a function?
The distance between the points (2,1) and (x,7) as graphed on the standard (x,y) coordinate plane is 10. What is one possible value for x?
A. -10
B. -6
C. 5
D. 8
Could you repost answer explanation to CollegeBoard Test 3, Math 4.23? Specifically, is there a way to solve this if you don’t know the little trig ID about complementary angles?
2x^2 – 3x – 7 = 0
If c and d are the two solutions of the quadratic equation above, what is the value of c + d?
Is there a smarter/quicker way to solve this than the quadratic formula?
Which of the following equations describes a circle with radius 10 that passes through the origin when graphed in the xy-plane?
A) (x – 5)² + (y+5)² = 10
B) (x – 5)² + (y+5)² = 100
C) (x – 10)² + (y+10)² = 10
D) (x – 5√2)² + (y+5√2)² = 100
Clearly, A) is out because that one does not have a radius of 10. What is the most time-efficient way to solve this? Sketch and eyeball?
Can you explain this algebraically (even if you also give a graphed explanation)?
Radioactive substances decay over time. the mass M, in grams, of a particular radioactive substance d days after the beginning of an experiment is shown in the table below:
Number of days, d Mas, M (grams)
0 120.00
30 103.21
60 88.78
90 76.36
If this relationship is modeled by the function M(d) = a • 10^bd, which of the following could be the values of a and b?
A) a = 12 and b = 0.0145
B) a = 12 and b = -0.0145
C) a = 120 and b = 0.0022
D) a = 120 and b = -0.0022
College Board Test 4, Math 4 #25 (the explanation link that you previously made does not work):
f(x) = 2x^3 + 6 x^2 + 4x
g(x) = x^2 + 3x + 2
The polynomials f(x) and g(x) are defined above. Which of the following polynomials is divisible by 2x + 3?
A) h(x) = f(x) + g(x)
B) p(x) = f(x) + 3 g(x)
C) r(x) = 2 f(x) + 3 g(x)
D) s(x) = 3 f(x) + 2 g(x)
Woops! Forgot part of first equation in question just submitted. Here is the full question:
(x + 2)^2 + (y – 3)^2 = 40
y = -2x + 4
Which could be the x-coordinate of a solution to the system of equations above?
A) √7
B) √35 / 2
C) (6 + 2√34) /5
D) (4 + 2√191) /5
4x + y = 7
2x – 7y = 1
If I multiply the second equation by 2, I can stack them and subtract:
4x + y = 7
4x – 14y = 2
So, 15y = 5, —> y = 3
Then: 2x – 7(3) = 1 –> 2x – 21 = 3 —> 2x = 24 –> x = 12
But: 4x + 3 = 7 –> 4x = 4 –> x = 1
What am I not seeing? The answer should be x= 5/3.
The lengths of the sides of a rectangle are a and b, where a > b The sum of the lengths of the two shorter sides and one of the longer sides of the rectangle is 36. What value of a maximizes the area of the rectangle?
A .9
B. 12
C. 18
D. 24
The answer is C. I suspect there is an easier way to solve than completing the square and finding the vertex of the resulting quadratic function. What is your most direct, easy to understand solution to this calculator-allowed question?
g(x) = 1/5 (5)^(x+4)
For the given function g, which of the following equivalent forms shows the y-coordinate of the y-intercept of the graph of y=g(x) in the xy-plane as a constant or coefficient?
A. g(x) = 125(5)^x
B. g(x) = 25(5)^(x+1)
C. g(x) = 5(5)^(x+2)
D. g(x) = (5)^(x+3)
Choice A is correct, but why?
x f(x)
____________
-5 0
-1 36
0 0
1 6
2 0
The function f is defined by a polynomial. Some values of x and f(x) are shown in the table above. Which of the following could define f?
A) (x – 5) (x + 2)
B) (x + 5)^2 (x – 2)^3
C) x^2 (x + 5) (x – 2)^2
D) x (x + 5) (x – 2)
Specifically, can you explain why C is correct but D cannot be?
In a circle with area of 120 to 124 sq inches, the area of the sector formed by an angle is between 20 and 21 sq inches. What is one possible integer value of the angle?
I came up with a low value of 60 (if area of circle is on the lowest end, 120 sq inches, and area of the sector is also on the low end, 20 inches). If both those areas are on the highest end, then I came up with 60 again. But answer is supposed to be 59 ≤ x ≤ 63. Does this make sense, and if so, can you explain it?
Can you explain a more direct way to solve College Board Official Practice Test 9, Math Section 4 #19, than the College Board’s explanation? I seem to remember something about making a chart to solve mixture problems. Would that work here?
