Question

1. determine the degree of each polynomial function represented in the tables below.

$x_1$	$f(x_1)$		$x_1$	$g(x_1)$
-2	-4		-2	6
-1	1		-1	0
0	0		0	2
1	-1		1	0
2	4		2	6
3	21		3	56
4	56		4	210
			5	552

1. consider $k(x) = 3(x - 1)(x + 8)(8x - 5)(6x + 21)$.

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

Explanation:

Problem 2 (Determine the degree of polynomial functions from tables)

For \( f(x_1) \):

Step 1: Calculate finite differences

First, list the \( x \) and \( f(x) \) values: \( x = [-3, -2, -1, 0, 1, 2, 3, 4] \), \( f(x) = [-21, -4, 1, 0, -1, 4, 21, 56] \).
Calculate first differences (Δ1):
\( -4 - (-21) = 17 \), \( 1 - (-4) = 5 \), \( 0 - 1 = -1 \), \( -1 - 0 = -1 \), \( 4 - (-1) = 5 \), \( 21 - 4 = 17 \), \( 56 - 21 = 35 \).
Second differences (Δ2):
\( 5 - 17 = -12 \), \( -1 - 5 = -6 \), \( -1 - (-1) = 0 \), \( 5 - (-1) = 6 \), \( 17 - 5 = 12 \), \( 35 - 17 = 18 \).
Third differences (Δ3):
\( -6 - (-12) = 6 \), \( 0 - (-6) = 6 \), \( 6 - 0 = 6 \), \( 12 - 6 = 6 \), \( 18 - 12 = 6 \).
The third differences are constant (\( 6 \)), so the degree of \( f(x) \) is \( 3 \).

Step 2: For \( g(x_1) \)

List \( x = [-3, -2, -1, 0, 1, 2, 3, 4, 5] \), \( g(x) = [56, 6, 0, 2, 0, 6, 56, 210, 552] \).
First differences (Δ1):
\( 6 - 56 = -50 \), \( 0 - 6 = -6 \), \( 2 - 0 = 2 \), \( 0 - 2 = -2 \), \( 6 - 0 = 6 \), \( 56 - 6 = 50 \), \( 210 - 56 = 154 \), \( 552 - 210 = 342 \).
Second differences (Δ2):
\( -6 - (-50) = 44 \), \( 2 - (-6) = 8 \), \( -2 - 2 = -4 \), \( 6 - (-2) = 8 \), \( 50 - 6 = 44 \), \( 154 - 50 = 104 \), \( 342 - 154 = 188 \).
Third differences (Δ3):
\( 8 - 44 = -36 \), \( -4 - 8 = -12 \), \( 8 - (-4) = 12 \), \( 44 - 8 = 36 \), \( 104 - 44 = 60 \), \( 188 - 104 = 84 \).
Fourth differences (Δ4):
\( -12 - (-36) = 24 \), \( 12 - (-12) = 24 \), \( 36 - 12 = 24 \), \( 60 - 36 = 24 \), \( 84 - 60 = 24 \).
The fourth differences are constant (\( 24 \)), so the degree of \( g(x) \) is \( 4 \).

Problem 3 (Find \( x \)-intercepts and \( y \)-intercept of \( k(x) = 3(x - 1)(x + 8)(8x - 5)(6x + 21) \))

Step 1: Find \( x \)-intercepts

\( x \)-intercepts occur where \( k(x) = 0 \). Set each factor to zero:

• \( x - 1 = 0 \implies x = 1 \)
• \( x + 8 = 0 \implies x = -8 \)
• \( 8x - 5 = 0 \implies x = \frac{5}{8} \)
• \( 6x + 21 = 0 \implies 6x = -21 \implies x = -\frac{21}{6} = -\frac{7}{2} \)

Step 2: Find \( y \)-intercept

\( y \)-intercept occurs at \( x = 0 \). Substitute \( x = 0 \) into \( k(x) \):
\( k(0) = 3(0 - 1)(0 + 8)(8(0) - 5)(6(0) + 21) \)
\( = 3(-1)(8)(-5)(21) \)
\( = 3 \times (-1) \times 8 \times (-5) \times 21 \)
\( = 3 \times 840 \)
\( = 2520 \)

Answer:

s:

Problem 2:

• Degree of \( f(x_1) \): \( \boldsymbol{3} \)
• Degree of \( g(x_1) \): \( \boldsymbol{4} \)

Problem 3:

• \( x \)-intercepts: \( \boldsymbol{1, -8, \frac{5}{8}, -\frac{7}{2}} \)
• \( y \)-intercept: \( \boldsymbol{2520} \)