1. Use the factor theorem to determine whether the expressions are factors of the polynomial P(x). a. P(x) = x^5 + 243; x + 3 b. P(x) = x^3 + 2x^2 - x - 2; x - 1 c. P(x) = x^3 - 3x... 1. Use the factor theorem to determine whether the expressions are factors of the polynomial P(x). a. P(x) = x^5 + 243; x + 3 b. P(x) = x^3 + 2x^2 - x - 2; x - 1 c. P(x) = x^3 - 3x^2 - 4x - 12; x + 2 2. Find the value of k. a. For which x^3 - 2x^2 + 4x + k has a remainder of -7 when divided by x - 1 b. So that x + 2 is a factor of P(x) = 3x^3 + 5x^2 + kx - 10 3. A specific car's fuel economy in miles per gallon can be approximately given by f(x) = 0.0000056x^4 - 0.000018x^3 + 1.38x - 0.38, where x represents the car's speed in miles per hour. Determine the fuel economy when the car is travelling 40, 50, and 60 miles per hour. a. f(40) b. f(50)? c. f(60)? d. Which car speed maximizes fuel consumption? BONUS: ALL or NOTHING Determine whether x - b is a factor of x^3 - (2a + b)x^2 + (3a + 2ab)x - 3ab. Read more

Steps to Solve

1. Use the Factor Theorem for Expression a
   To check if $x + 3$ is a factor of $P(x) = x^5 + 243$, substitute $x = -3$ into $P(x)$:
   $$ P(-3) = (-3)^5 + 243 = -243 + 243 = 0. $$
   Since the result is $0$, $x + 3$ is a factor.

2. Use the Factor Theorem for Expression b
   To check if $x - 1$ is a factor of $P(x) = x^3 + 2x^2 - x - 2$, substitute $x = 1$:
   $$ P(1) = 1^3 + 2(1^2) - 1 - 2 = 1 + 2 - 1 - 2 = 0. $$
   Since the result is $0$, $x - 1$ is a factor.

3. Use the Factor Theorem for Expression c
   To check if $x + 2$ is a factor of $P(x) = x^3 - 3x^2 - 4x - 12$, substitute $x = -2$:
   $$ P(-2) = (-2)^3 - 3(-2)^2 - 4(-2) - 12 = -8 - 12 + 8 - 12 = -24. $$
   Since the result is not $0$, $x + 2$ is not a factor.

4. Finding k for Expression a
   For the polynomial $x^3 - 2x^2 + 4x + k$, we need it to have a remainder of $-7$ when divided by $x - 1$:
   $$ P(1) = 1^3 - 2(1)^2 + 4(1) + k = 1 - 2 + 4 + k = 3 + k. $$
   Set this equal to $-7$:
   $$ 3 + k = -7 \implies k = -10. $$

5. Finding k for Expression b
   For $x + 2$ to be a factor of $P(x) = 3x^3 + 5x^2 + kx - 10$, substitute $x = -2$:
   $$ P(-2) = 3(-2)^3 + 5(-2)^2 + k(-2) - 10. $$
   Calculate:
   $$ P(-2) = 3(-8) + 5(4) - 2k - 10 = -24 + 20 - 2k - 10 = -14 - 2k. $$
   Set this equal to $0$:
   $$ -14 - 2k = 0 \implies 2k = -14 \implies k = -7. $$

6. Calculating Fuel Economy for Different Speeds
   Fuel economy function:
   $$ f(x) = 0.00000056x^4 - 0.000018x^3 + 1.38x - 0.38. $$
   Substitute $x = 40, 50, 60$ respectively into the function:

   • For $f(40)$: $$ f(40) = 0.00000056(40)^4 - 0.000018(40)^3 + 1.38(40) - 0.38. $$
   • For $f(50)$:
     $$ f(50) = 0.00000056(50)^4 - 0.000018(50)^3 + 1.38(50) - 0.38. $$
   • For $f(60)$:
     $$ f(60) = 0.00000056(60)^4 - 0.000018(60)^3 + 1.38(60) - 0.38. $$

7. Maximizing Fuel Consumption
   To find which speed maximizes fuel consumption, analyze $f(x)$ over the interval of interest (suitable derivative and testing critical points).