Question

6.)

1. $(f + 1)(f^2 + 4f + 8)$
2. $(t - 2)(t^2 - 5t + 1)$
3. $(6 + d)(2d^2 - d + 7)$

Explanation:

Problem 48: \((f + 1)(f^2 + 4f + 8)\)

Step1: Distribute \(f\)

Multiply \(f\) with each term in \(f^2 + 4f + 8\):
\(f \cdot f^2 + f \cdot 4f + f \cdot 8 = f^3 + 4f^2 + 8f\)

Step2: Distribute \(1\)

Multiply \(1\) with each term in \(f^2 + 4f + 8\):
\(1 \cdot f^2 + 1 \cdot 4f + 1 \cdot 8 = f^2 + 4f + 8\)

Step3: Combine like terms

Add the results from Step1 and Step2:
\(f^3 + 4f^2 + 8f + f^2 + 4f + 8 = f^3 + (4f^2 + f^2) + (8f + 4f) + 8 = f^3 + 5f^2 + 12f + 8\)

Problem 50: \((t - 2)(t^2 - 5t + 1)\)

Step1: Distribute \(t\)

Multiply \(t\) with each term in \(t^2 - 5t + 1\):
\(t \cdot t^2 + t \cdot (-5t) + t \cdot 1 = t^3 - 5t^2 + t\)

Step2: Distribute \(-2\)

Multiply \(-2\) with each term in \(t^2 - 5t + 1\):
\(-2 \cdot t^2 + (-2) \cdot (-5t) + (-2) \cdot 1 = -2t^2 + 10t - 2\)

Step3: Combine like terms

Add the results from Step1 and Step2:
\(t^3 - 5t^2 + t - 2t^2 + 10t - 2 = t^3 + (-5t^2 - 2t^2) + (t + 10t) - 2 = t^3 - 7t^2 + 11t - 2\)

Problem 52: \((6 + d)(2d^2 - d + 7)\)

Step1: Distribute \(6\)

Multiply \(6\) with each term in \(2d^2 - d + 7\):
\(6 \cdot 2d^2 + 6 \cdot (-d) + 6 \cdot 7 = 12d^2 - 6d + 42\)

Step2: Distribute \(d\)

Multiply \(d\) with each term in \(2d^2 - d + 7\):
\(d \cdot 2d^2 + d \cdot (-d) + d \cdot 7 = 2d^3 - d^2 + 7d\)

Step3: Combine like terms

Add the results from Step1 and Step2:
\(2d^3 - d^2 + 7d + 12d^2 - 6d + 42 = 2d^3 + (-d^2 + 12d^2) + (7d - 6d) + 42 = 2d^3 + 11d^2 + d + 42\)

Answer:

s:

1. \(\boldsymbol{f^3 + 5f^2 + 12f + 8}\)
2. \(\boldsymbol{t^3 - 7t^2 + 11t - 2}\)
3. \(\boldsymbol{2d^3 + 11d^2 + d + 42}\)