Question

unit 3: polynomial functions
3.1.a independent practice & hw

1. convert $f(x) = 3(x - 5)(x + 7)(x - 1)$ to standard form.

$x$-intercepts:
$y$-intercept:

1. convert $g(x) = - 3(x + 8)(x - 2)(x - 4)$ to standard form.

$x$-intercepts:
$y$-intercept:

1. consider $h(x) = - 5(x - 6)(x + 2)(x + 4)(x - 11)$.

$x$-intercepts:
$y$-intercept:

1. consider $k(x) = 3(x - 3)(x + 12)(4x - 3)(2x + 5)$.

$x$-intercepts:
$y$-intercept:

1. consider $k(x) = - 2(x - 1)(x + 5)(7x - 2)(5x + 8)$.

$x$-intercepts:
$y$-intercept:

Answer:

Problem 1: Convert \( f(x) = 3(x - 5)(x + 7)(x - 1) \) to standard form, find \( x \)-intercepts and \( y \)-intercept.

Step 1: Find \( x \)-intercepts

To find \( x \)-intercepts, set \( f(x) = 0 \):

$$ 3(x - 5)(x + 7)(x - 1) = 0 $$

Using the zero - product property (if \( ab = 0\), then either \( a = 0\) or \( b = 0\)), we have:
\( x - 5=0\) or \( x + 7 = 0\) or \( x - 1=0\)
Solving these equations:

• For \( x - 5=0\), we get \( x = 5\)
• For \( x + 7=0\), we get \( x=-7\)
• For \( x - 1=0\), we get \( x = 1\)

So the \( x \)-intercepts are \( x = 5\), \( x=-7\), \( x = 1\)

Step 2: Expand the function to standard form

First, multiply \( (x - 5)(x + 7) \):

$$
LATEXBLOCK0
$$

Then multiply the result by \( (x - 1) \):

$$
LATEXBLOCK1
$$

Now multiply by 3:
\( f(x)=3(x^{3}+x^{2}-37x + 35)=3x^{3}+3x^{2}-111x + 105\)

Step 3: Find \( y \)-intercept

To find the \( y \)-intercept, set \( x = 0\) in \( f(x) \):

$$
LATEXBLOCK2
$$

Problem 2: Convert \( g(x)=-3(x + 8)(x - 2)(x - 4) \) to standard form, find \( x \)-intercepts and \( y \)-intercept.

Step 1: Find \( x \)-intercepts

Set \( g(x)=0\):

$$ -3(x + 8)(x - 2)(x - 4)=0 $$

Using the zero - product property:
\( x + 8=0\) or \( x - 2=0\) or \( x - 4=0\)

• For \( x + 8=0\), \( x=-8\)
• For \( x - 2=0\), \( x = 2\)
• For \( x - 4=0\), \( x = 4\)

So the \( x \)-intercepts are \( x=-8\), \( x = 2\), \( x = 4\)

Step 2: Expand the function

First, multiply \( (x + 8)(x - 2) \):

$$
LATEXBLOCK3
$$

Then multiply by \( (x - 4) \):

$$
LATEXBLOCK4
$$

Multiply by - 3:
\( g(x)=-3(x^{3}+2x^{2}-40x + 64)=-3x^{3}-6x^{2}+120x - 192\)

Step 3: Find \( y \)-intercept

Set \( x = 0\) in \( g(x) \):

$$
LATEXBLOCK5
$$

Problem 3: For \( h(x)=-5(x - 6)(x + 2)(x + 4)(x - 11) \), find \( x \)-intercepts and \( y \)-intercept.

Step 1: Find \( x \)-intercepts

Set \( h(x)=0\):

$$ -5(x - 6)(x + 2)(x + 4)(x - 11)=0 $$

Using the zero - product property:
\( x - 6=0\) or \( x + 2=0\) or \( x + 4=0\) or \( x - 11=0\)

• \( x = 6\)
• \( x=-2\)
• \( x=-4\)
• \( x = 11\)

So the \( x \)-intercepts are \( x = 6\), \( x=-2\), \( x=-4\), \( x = 11\)

Step 2: Find \( y \)-intercept

Set \( x = 0\) in \( h(x) \):

$$
LATEXBLOCK6
$$

Problem 4: For \( k(x)=3(x - 3)(x + 12)(4x - 3)(2x + 5) \), find \( x \)-intercepts and \( y \)-intercept.

Step 1: Find \( x \)-intercepts

Set \( k(x)=0\):

$$ 3(x - 3)(x + 12)(4x - 3)(2x + 5)=0 $$

Using the zero - product property:
\( x - 3=0\) or \( x + 12=0\) or \( 4x - 3=0\) or \( 2x + 5=0\)

• \( x = 3\)
• \( x=-12\)
• \( 4x=3\Rightarrow x=\frac{3}{4}\)
• \( 2x=-5\Rightarrow x=-\frac{5}{2}\)

So the \( x \)-intercepts are \( x = 3\), \( x=-12\), \( x=\frac{3}{4}\), \( x=-\frac{5}{2}\)

Step 2: Find \( y \)-intercept

Set \( x = 0\) in \( k(x) \):

$$
LATEXBLOCK7
$$

Problem 5: For \( k(x)=-2(x - 1)(x + 5)(7x - 2)(5x + 8) \), find \( x \)-intercepts and \( y \)-intercept.

Step 1: Find \( x \)-intercepts

Set \( k(x)=0\):

$$ -2(x - 1)(x + 5)(7x - 2)(5x + 8)=0 $$

Using the zero - product property:
\( x - 1=0\) or \( x + 5=0\) or \( 7x - 2=0\) or \( 5x + 8=0\)

• \( x = 1\)
• \( x=-5\)
• \( 7x=2\Rightarrow x=\frac{2}{7}\)
• \( 5x=-8\Rightarrow x=-\frac{8}{5}\)

So the \( x \)-intercepts are \( x = 1\), \( x=-5\), \( x=\frac{2}{7}\), \( x=-\frac{8}{5}\)

Step 2: Find \( y \)-intercept

Set \( x = 0\) in \( k(x) \):

$$
LATEXBLOCK8
$$

Final Answers

Problem 1

• \( x \)-intercepts: \( x = 5\), \( x=-7\), \( x = 1\)
• Standard form: \( f(x)=3x^{3}+3x^{2}-111x + 105\)
• \( y \)-intercept: \( y = 105\)

Problem 2

• \( x \)-intercepts: \( x=-8\), \( x = 2\), \( x = 4\)
• Standard form: \( g(x)=-3x^{3}-6x^{2}+120x - 192\)
• \( y \)-intercept: \( y=-192\)

Problem 3

• \( x \)-intercepts: \( x = 6\), \( x=-2\), \( x=-4\), \( x = 11\)
• \( y \)-intercept: \( y=-2640\)

Problem 4

• \( x \)-intercepts: \( x = 3\), \( x=-12\), \( x=\frac{3}{4}\), \( x=-\frac{5}{2}\)
• \( y \)-intercept: \( y = 1620\)

Problem 5

• \( x \)-intercepts: \( x = 1\), \( x=-5\), \( x=\frac{2}{7}\), \( x=-\frac{8}{5}\)
• \( y \)-intercept: \( y=-160\)