Quadratics, polynomials, and complex logic. The final barrier to an 800.
What is the remainder when \(f(x) = 2x^{20} - 5x^3 + 4\) is divided by \(x - 1\)?
Trying to perform long division. An \(x^{20}\) term will steal 5 minutes and grant you 0 points.
Use the Remainder Theorem. The remainder is simply \(f(1)\).
2(1) - 5(1) + 4 = 1
Answer: 1. Instant victory.
\(x^2 + kx + 9 = 0\) has exactly one real solution. If \(k > 0\), find \(k\).
Guessing for x or k. This is a purely conceptual "discriminant" check.
One solution means \(b^2 - 4ac = 0\).
k² - 36 = 0 \(\rightarrow\) k = 6
Answer: 6.
Equivalent form of \(\frac{3 + i}{2 - i}\)?
Treating it like a simple fraction. You must eliminate the complex denominator.
Multiply by the conjugate \((2 + i)\).
\(\frac{5+5i}{5} = 1 + i\)
Answer: 1 + i.
\(y = x^2 - 6x + 5\). Convert to vertex form.
Completing the square manually. Slow.
\(h = -b/2a = 6/2 = 3\). Plug in 3: \(3^2 - 18 + 5 = -4\). Vertex (3, -4).
Answer: \(y = (x-3)^2 - 4\).
\(2x^2 + bx + 8 = 0\) has 2 distinct real solutions. Condition for b?
Using \(\ge 0\). Distinct means STRICTLY greater than 0.
\(b^2 - 4(2)(8) > 0\). \(b^2 - 64 > 0\). \(|b| > 8\). So \(b > 8\) or \(b < -8\).
Answer: \(|b| > 8\).
\(x^2 + 4x + c = 0\) has no real solutions. Range for c?
Forgetting the inequality direction.
\(b^2 - 4ac < 0\). \(16 - 4c < 0\). \(16 < 4c\). \(4 < c\).
Answer: \(c > 4\).
\(3x^2 - 12x + 7 = 0\). What is value of \(x_1 + x_2\)?
Solving for the roots with quadratic formula. Painful.
Sum = \(-b/a\). \(-(-12)/3 = 4\). Done.
Answer: 4.
\(2x^2 + 5x - 8 = 0\). Find \(x_1 \cdot x_2\).
Again, trying to solve properly.
Product = \(c/a\). \(-8/2 = -4\).
Answer: -4.
If \(f(-3) = 0\), what must be a factor of \(f(x)\)?
Thinking \(x-3\). Signs flip.
Root at -3 means factor is \(x - (-3)\), which is \(x+3\).
Answer: \(x+3\).
\(\frac{4x+2}{x+2}\) is equivalent to \(4 - \frac{A}{x+2}\). Find A.
Algebraic long division. Just plug in a number!
Let \(x=0\). LHS: \(2/2=1\). RHS: \(4 - A/2\). So \(1 = 4 - A/2\). \(A/2 = 3\). \(A = 6\).
Answer: 6.
Element decays by half every 3 years. P(t) = \(100(0.5)^{t/k}\). Find k.
Thinking k is related to the rate. k is the time period.
In the formula \(a \cdot r^{t/h}\), h is the "halving time". Here, it's 3.
Answer: 3.
\(\sqrt{2x+6} = x-1\). How many valid solutions?
Solving and forgetting to check back in. Squaring creates fake roots.
\(2x+6 = (x-1)^2\). \(x^2 - 4x - 5 = 0\). \((x-5)(x+1)\). Roots: 5, -1. Check -1: \(\sqrt{4} = -2\)? No. Only 5 works.
Answer: 1.
\(\frac{1}{x-2} + \frac{1}{x+3} = 5\). What value of x is impossible?
Trying to solve for x. Just look at the bottom.
Denominator cannot be zero. \(x \ne 2\) and \(x \ne -3\).
Answer: 2 and -3.