HW 5: Polynomial Inequalities, Roots, and Symmetry

Vocabulary practice flashcards covering polynomial inequalities, real and complex roots, x-intercepts, and function symmetry based on HW 5 lecture notes.

Solution set to $(2x+6)^4(x-7)^3 \le 0$

$$(-\infty, 7]$$

Solution set to $3x^3(2x-6) < 0$

$$(0, 3)$$

Solution set to $8x^2(3x+6)^3 < 0$

$$(-\infty, -2)$$

Solution set to $(x-8)(-7x-9) < 0$

$$(-\infty, -\frac{9}{7}) \cup (8, \infty)$$

Solution set to $6x^2(3x+6) \ge 0$

$$[-2, \infty)$$

Solution set to $-9x^3(7x+9)^3 > 0$

$$(-\frac{9}{7}, 0)$$

Real roots of $f(x) = x(x+6)(x+4)$

$0$, $-6$, and $-4$

Real roots of $f(x) = (x-8)(x+1)(x-5)$

$8$, $-1$, and $5$

Real roots of $f(x) = (x-7)(x^2-1)$

$7$, $-1$, and $1$

Real roots of $f(x) = (x-6)(x^2-4)$

$6$, $-2$, and $2$

Real roots of $f(x) = x(x^2-25)$

$0$, $-5$, and $5$

Roots of $f(x) = (x-3)(x^2+1)$

Real root $x = 3$ only (complex roots $x = \pm i$)

Roots of $f(x) = (x+6)(x^2+4)$

Real root $x = -6$ only (complex roots $x = \pm 2i$)

Real roots of $f(x) = (x+6)(x^2-12)$

$-6$, $3.5$, and $-3.5$ (where $x \approx \pm 3.5$)

Real roots of $f(x) = (x+5)(x^2-7)$

$-5$, $2.6$, and $-2.6$ (where $x \approx \pm 2.6$)

$x$-intercepts of $f(x) = (x+2)^2(x+4)$

$x = -2$ and $x = -4$

$x$-intercepts of $f(x) = (x+8)(x+1)^2$

$x = -8$ and $x = -1$

$x$-intercepts of $f(x) = (x-6)^2(x-4)^2$

$x = 6$ and $x = 4$

Symmetry classification of $f(x) = 9x^3 + 8$

Neither even nor odd ($f(-x) \neq f(x)$ and $-f(x) \neq f(-x)$)

Symmetry classification of $f(x) = -x^2 + 6x^4 + 1$

Even function ($f(-x) = f(x)$)

Symmetry classification of $f(x) = -6x^7 - x^5$

Odd function ($-f(x) = f(-x)$)

Symmetry classification of $f(x) = -3x^3 + 3x^7 + x$

Odd function ($-f(x) = f(-x)$)

Graphical properties of an even degree, even function

End arrows point in the same direction, and the graph is symmetric about the y-axis