A question on polynomials practice question #10 (p. 149 in PWN the SAT Math Guide): I understand how to solve using polynomial long division but can you explain your shortcut from the answer explanation? How would we know to plug in -3?
Could you further explain your explanation of Practice Question #4 in PWN the SAT Math Guide “Polynomials” chapter (p. 148). In your long division confirmation of the answer (explanation p. 326), how do you know to divide by 9x + 1?
Can you suggest a quick method to solve College Board Practice Test 5 Math Section 3 #14? I plugged in easy numbers for x in order to eliminate answer choices (first x=1, then x=0), but wonder if there is a more direct way to solve.
In the “Proving Grounds” Quiz 1 #5, this question is given:
p(x) = ax^2 + bx + c
q(x) = ax^2 + bx + d
The functions p(x) and q(x) are defined above, and a, b, c, and d are constants. If x – 3 is a factor of p(x) and d is 8 greater than c, what is the remainder when q(x) is divided by x – 3?
In your answer explanation, you state:
“…We know that, as a general rule, the remainder when a polynomial f(x) is divided by x – a is f(a).”
Can you explain this further?
Is there a way to solve this system of equations without using the quadratic formula (or graphing)?
f(x) = –2/3 x + 4
g(x) = 3(x + 2)^2 – 4
How many solutions does the system above have?
y > -½ x + a
y > 2 x – a
If the value of a is 7, which of the following is a possible solution to the system of inequalities above?
A) (2, 6)
B) (4, 5)
C) (5, 4)
D) (5, 3)
Is there a better way than plugging in the answers to solve this?
f(x) = √x
g(x) = 3x – b
If the graph of y = f(g(x)) passes through (6, 5) in the standard (x, y) coordinate plane, what is the value of b?
a = 4800 – 6t
b = 5400 – 8t
In the system of equations above, a and b represent the distance, in meters, two marathon runners are from the finish line after running for four hours and t seconds. How far will runner a be from the finish line when runner b passes her?
A. 300 meters
B. 500 meters
C. 100 meters
D. 3000 meters
A rideshare app charges $2 per trip plus $0.4 per mile. A competitor charges $1 for the first 6 miles plus $0.5 per mile for every additional mile. For what length trip would the two services charge the same amount?
A. 10 miles
B. 18 miles
C. 20 miles
D. 40 miles
Can you craft an algebraic equation to solve this directly – or is plugging in the answers the way to go?
This may be a little advanced for the SAT, but complex numbers sometimes show up –as do cubic polynomials– so hopefully you can address this for me! TIA!
Which of the following could be the full set of complex roots of a cubic polynomial with real coefficients?
A. { 0, 1, i}
B. {1, i, 2i}
C. {2, i}
D. {3, 2 + i, 2 – i}
A question about composite functions:
If f(x) = √x and g(x) = x^2, and you are solving for f(g(x)), ordinarily you solve for g(x) and then plug that value into f(x) to solve for f(g(x)). But what if x is a negative number? When you square it, you’ll get a positive value and then when you take the square root of that to solve for g(f(x)), the final answer will be positive only. Is that correct?
A logarathim question:
pH = – log h
The pH of a solution is dependent on the concentration of hydronium ions, h, and can be calculated by the equation above. If the concentration of hydronium in solution A is 100 times the concentration of the hydronium in solution B, what is the absolute value of the difference in their pH values?
A. –2
B. 2
C. 3
D. 10
E. 100
One gallon of honey weighs approximately 12 lbs. If one gallon of honey is mixed with 5 gallons of water to make tea, how many ounces of honey will be in each 8 fluid ounce cup of tea?
Choices are 1, 2, 3 or 4. (Answer= 2) given: 16 oz = 1 lb, 128 fl oz = 1 gallon
PWN the SAT Parabolas drill explanation p. 325 #10: The final way to solve: If we are seeking x=y, since the point is (a,a), why can you set f(x) = 0? You start out with the original equation in vertex form, making y=a and x=a, but halfway through you change to y=0 (while x is still = a). How can we be solving the equation when we no longer have a for both x and y?
A triangle’s base was increased by 15%. If its area is increased by 38%, what percent was the height of the triangle increased by?