Question

question 7 (mandatory) (1 point) perform the indicated operations and simplify. \\((x^2 + x - 2)(x^3 - x + 8)\\) \\(\circ\\) a) \\(x^5 + x^4 + 3x^3 - 7x^2 - x - 8\\) \\(\circ\\) b) \\(x^5 + x^4 - 3x^3 + 7x^2 + 10x - 16\\) \\(\circ\\) c) \\(x^5 - x^4 + 3x^3 - 7x^2 + 10x - 16\\) \\(\circ\\) d) \\(x^5 - 7x^4 - 3x^3 + x^2 - 10x + 2\\) \\(\circ\\) e) \\(x^5 + 3x^4 - x^3 + 7x^2 - 10x + 16\\)

Explanation:

Step1: Distribute \(x^2\)

Multiply \(x^2\) with each term in \((x^3 - x + 8)\):
\(x^2 \cdot x^3 = x^5\), \(x^2 \cdot (-x) = -x^3\), \(x^2 \cdot 8 = 8x^2\)

Step2: Distribute \(x\)

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

Step3: Distribute \(-2\)

Multiply \(-2\) with each term in \((x^3 - x + 8)\):
\(-2 \cdot x^3 = -2x^3\), \(-2 \cdot (-x) = 2x\), \(-2 \cdot 8 = -16\)

Step4: Combine like terms

• \(x^5\) (only term)
• \(x^4\) (only term)
• \(x^3\): \(-x^3 - 2x^3 = -3x^3\)? Wait, no—wait, original steps: Wait, re - check:

Wait, from Step1: \(x^5 - x^3 + 8x^2\)
Step2: \(x^4 - x^2 + 8x\)
Step3: \(-2x^3 + 2x - 16\)

Now combine:
\(x^5 + x^4 + (-x^3 - 2x^3)+(8x^2 - x^2)+(8x + 2x)-16\)
Simplify each group:

• \(x^5\)
• \(x^4\)
• \(x^3\): \(-3x^3\)? Wait, no, wait the original problem is \((x^2 + x - 2)(x^3 - x + 8)\). Wait, maybe I made a mistake in distribution. Let's do it again:

\((x^2 + x - 2)(x^3 - x + 8)\)
\(= x^2(x^3 - x + 8)+x(x^3 - x + 8)-2(x^3 - x + 8)\)
\(= x^5 - x^3 + 8x^2 + x^4 - x^2 + 8x - 2x^3 + 2x - 16\)

Now combine like terms:

• \(x^5\)
• \(x^4\)
• \(x^3\): \(-x^3 - 2x^3=-3x^3\)? Wait, no, wait the option b is \(x^5 + x^4 - 3x^3 + 7x^2 + 10x - 16\). Let's check \(x^2\) terms: \(8x^2 - x^2 = 7x^2\)

\(x\) terms: \(8x + 2x = 10x\)
Constants: \(-16\)

So putting it all together: \(x^5 + x^4 - 3x^3 + 7x^2 + 10x - 16\)

Answer:

b) \(x^5 + x^4 - 3x^3 + 7x^2 + 10x - 16\)