Question

warm up log
quote of the week
\practice isnt the thing you do once youre good. its the thing you do that makes you good.\
monday:
$f(x) = x(x - 1)(x - 2)(x - 3)$
tuesday:
if the population of a certain species doubles every nine years. if the population started with 100 individuals, which of the following expressions gives the population of the species t years after the population started, assuming that the population has been living ideal conditions?
a) $2 \times 100^{9t}$ c) $100 \times 2^{9t}$
b) $2 \times 100^{t/9}$ d) $100 \times 2^{t/9}$
wednesday:
$f(x) = -x(x + 1)(x + 2)(x + 3)$
thursday:
which of the following is equivalent to the following expression?
$4(x + 5) + 4x + 8$
a) $4(2x + 7)$
b) $8(x + 4)$
c) $5x + 17$
d) $8x + 13$
$f(x) = -x^2(x + 3)(x - 3)$

Explanation:

Monday Step1: Identify roots of the polynomial

Set $f(x)=0$, solve $x(x-1)(x-2)(x-3)=0$. Roots are $x=0,1,2,3$.

Monday Step2: Determine end behavior

Degree is 4 (even), leading coefficient positive. As $x\to\pm\infty$, $f(x)\to+\infty$.

Tuesday Step1: Recall exponential growth formula

The general form is $P(t)=P_0 \times r^{\frac{t}{T}}$, where $P_0$ is initial population, $r$ is growth factor, $T$ is doubling time.

Tuesday Step2: Substitute given values

$P_0=100$, $r=2$, $T=9$. So $P(t)=100\times2^{\frac{t}{9}}$.

Wednesday Step1: Identify roots of the polynomial

Set $f(x)=0$, solve $-x(x+1)(x+2)(x+3)=0$. Roots are $x=0,-1,-2,-3$.

Wednesday Step2: Determine end behavior

Degree is 4 (even), leading coefficient negative. As $x\to\pm\infty$, $f(x)\to-\infty$.

Thursday Step1: Expand the expression

$4(x+5)+4x+8 = 4x+20+4x+8$

Thursday Step2: Combine like terms

$4x+4x+20+8=8x+28=8(x+4)$

Thursday (bottom) Step1: Simplify the polynomial

Use difference of squares: $(x+3)(x-3)=x^2-9$. So $f(x)=-x^2(x^2-9)=-x^4+9x^2$.

Thursday (bottom) Step2: Identify roots

Set $f(x)=0$, solve $-x^2(x^2-9)=0$. Roots are $x=0$ (double root), $x=3,-3$.

Thursday (bottom) Step3: Determine end behavior

Degree is 4 (even), leading coefficient negative. As $x\to\pm\infty$, $f(x)\to-\infty$.

Answer:

1. Monday: The polynomial has roots at $x=0,1,2,3$, opens upwards at both ends, and crosses the x-axis at each root.
2. Tuesday: d) $100 \times 2^{t/9}$
3. Wednesday: The polynomial has roots at $x=0,-1,-2,-3$, opens downwards at both ends, and crosses the x-axis at each root.
4. Thursday (top): b) $8(x+4)$
5. Thursday (bottom): The polynomial simplifies to $f(x)=-x^4+9x^2$, has roots at $x=0$ (double root, touches x-axis), $x=3,-3$ (crosses x-axis), and opens downwards at both ends.