Math / Chem

Math

Chem

Chem 7.31 - 7.45

HW 3.2 Q1

$f(x) = 8x^5 + 4x^3$

Coefficients: $8, 4$
Exponents: $5, 3$

The function is polynomial, and the degree is $5$ .

HW 3.2 Q2

$g(x) -= 3^5 - \pi x^4 + \frac{1}{8}x^3$

Coefficients: $3, \pi, \frac{1}{8}$
Exponents: $5, 4, 3$

The function is polynomial and the degree is $5$ .

HW 3.2 Q3

$h(x) = 6x^5 + 5x^3 + \frac{1}{x}$

$= 6x^5 + 5x^3 + 1x^{-1}$

Coefficients: $6, 5, 1$ $\checkmark$
Exponents: $5, 3, -1$ $\times$

Function is not polymonial.

HW 3.2 Q4

$f(x) = \frac{x^2+7}{x^2}$

$= x^0 + 15x^{-2}$

Coefficients: $1, 15$ $\checkmark$
Exponents: $0, -2$ $\times$

Function is not polymonial.

HW 3.2 Q7

$f(x) = -x^6 + x^4$

Leading coefficient $a = -1$
Exponent $n = 4$ is even
$a < 0$ and $n$ is even, therefore, graph falls to the left and to the right.

HW 3.2 Q8

Expand $f(x) = (x+4)^4$

Square of Difference: $a^2 + 2ab + b^2$

$f(x) = (x^2 + 2x(4) + 4^2)(x^2 + 2x(4) + 4^2)$

$= (x^2 + 8x + 16)(x^2 + 8x + 16)$

$= x^2(x^2 + 8x + 16) + 8x(x^2 + 8x + 16) + 16(x^2 + 8x + 16)$

$= x^4 + 8x^3 + 16x^2 + 8x^3 + 64x^2 + 128x + 16x^2 + 128x + 256$

$= x^4 + 16x^3 + 96x^2 + 256x + 256$

Leading coefficient $a = 1$
Coefficient $n = 4$
$a > 0$ and $n$ is even, therefore, graph rises to the left and rises to the right.

HW 3.2 Q9

$f(x) = 3x^7 + 3x^6 + 4x^5 + 2$

Leading coefficient: $a = 3$
Exponent: $n = 7$

$a > 0$ , $n$ is odd. Graph falls to the left and rises to the right.

HW 3.2 Q10

$f(x) = 3x^5 + 4x^ 3 - x + 7$

Leading coefficient: $a = 3$
Exponent: $n = 5$

$a > 0$ , $n$ is odd. Graph falls to the left and rises to the right.

HW 3.2 Q11

$f(x) = -3x^4 + 8x^2 - 5x + 7$

Leading coefficient: $a = -3$
Exponent: $n = 4$

$a < 0$ , $n$ is even. Graph falls to the left and falls to the right.

HW 3.2 Q12

$f(x) = -6(x+3)(x+1)^2$

Solve for $x$ when $f(x) = 0$
$-6(x+3)(x+1)^2 = 0$

Square of Difference: $a^2 + 2ab + b^2$

$x+3=0$
$x = -3$

$(x+1)^2 = 0$
$x + 1 = 0$
$x = -1$

Multiplicity of $-3$ is $1$
Multiplicity of $-1$ is $2$

$1$ is odd. Therefore, the graph crosses the x-axis at the zero $-3$ .
$2$ is even. Therefore, the graph touches the x-axis at the zero $-1$ .

HW 3.2 Q13

$f(x) = 7(x+5)(x+3)^3$

$x+5=0$
$x = -5$
Multiplcity is $1$ .

$(x+3)(x+3)(x+3) = 0$
$x = -3$
Multiplicity is $3$ .

HW 3.2 Q14

$f(x) = x^3-6x^2+9x$

Common term is $x$ . Factor out.

$x(x^2-6x + 9) = 0$

$x(x-3)(x-3)=0$

$x = 0$
Multiplicity is $1$ .
$x = 3$
Multiplicity is $2$ .

HW 3.2 Q15

$f(x) = x^3 + 4x^2 - x - 4$

Leading coefficient $a=1$
Degree or exponent $n=3$

For $n$ odd and $a$ > 0 : Graph falls to the left and rises to the right
For $n$ odd and $a$ < 0 : Graph rises to the left and falls to the right
For $n$ even and $a$ > 0 : Graph rises to the left and rises to the right
For $n$ even and $a$ < 0 : Graph falls to the left and falls to the right

$a>0$
$n$ is odd

Graph falls to the left and rises to the right.

$f(x) = x^3 + 4x^2 - x - 4$

$ax^3 + bx^2 + cx + d = 0$

$x^3 + 4x^2 - x - 4 = 0$

Common factor of first set is $x^2$
Common factor of second set is $-1$

$\fbox{$x^2(x+4)$} \fbox{$-1(x+4)$} = 0$

Both terms contain the same factor ( $x+4$ ), so you can combine the factors together.

$(x^2-1)(x+4) = 0$

Factor $(x^2-1)$ :
$x^2-1 = (x+1)(x-1)$

$(x+1)(x-1)(x+4) = 0$

$x = -1, x = 1, x = -4$

Multiplicity of all zeros is $1$ , which is odd. An odd multiplicity crosses the x-axis.

To find the y-intercept, set $x=0$

$f(x) = 0^3 + 4(0)^2 - 0 - 4$
$= -4$

$y=-4$

Compare $f(x)$ , $f(-x)$ and $-f(x)$

If $f(-x)=f(x)$ , the graph of $f$ is symmetric with resepct to the y-axis. If $f(-x) = -f(x)$ , the graph is symmetric with respect to the origin.

$f(x) = x^3 + 4x^2 - x - 4$
$f(-x) = (-x)^3 + 4(-x)^2 - (-x) - 4$
$= -x^3 +4x^2 + x -4$

$f(x) \ne f(-x)$

$f(x) = x^3 + 4x^2 - x - 4$
$-f(x) = -(x^3 + 4x^2 - x - 4)$
$= -x^3 - 4x^2 + x + 4$

$= -x^3 +4x^2 + x -4 \ne -x^3 - 4x^2 + x + 4$
$f(-x) \ne -f(x)$

$f$ does not have y-axis symmetry nor origin symmetry.

Maximum number of turning points is 1 less that the degree of the polynomial.

$3-1=2$

To draw the graph, plot 4 $x,y$ coordinates using the 4-point cubic tool.

The x-intercepts occur at $x = -1, x = 1, x = -4$

$(-1,0), (1,0), (-4,0)$

The y-intercept is, where $x=0$ occurs at $y=-4$

$(0,-4)$