Question

1. when (4x⁵ + 11x⁴ - 6x² + 24) is subtracted from (5x⁵ + 11x⁴ + 9x² - 25), what will be the result?
2. when (8x⁶ + 17x⁴ - 20x³ - 27x² - 6) is subtracted from (6x⁶ - 12x⁴ + 5x³ - 7x + 9), what will be the result?

Explanation:

Problem 2

Step1: Set up the subtraction

We need to subtract \((4x^5 + 11x^4 - 6x^2 + 24)\) from \((5x^5 + 11x^4 + 9x^2 - 25)\). So the expression is \((5x^5 + 11x^4 + 9x^2 - 25)-(4x^5 + 11x^4 - 6x^2 + 24)\).

Step2: Distribute the negative sign

\(5x^5 + 11x^4 + 9x^2 - 25 - 4x^5 - 11x^4 + 6x^2 - 24\).

Step3: Combine like terms

• For \(x^5\) terms: \(5x^5-4x^5 = x^5\)
• For \(x^4\) terms: \(11x^4 - 11x^4=0\)
• For \(x^2\) terms: \(9x^2 + 6x^2=15x^2\)
• For constant terms: \(-25-24=-49\)

So the result is \(x^5 + 15x^2 - 49\).

Problem 3

Step1: Set up the subtraction

We need to subtract \((8x^6 + 17x^4 - 20x^3 - 27x^2 - 6)\) from \((6x^6 - 12x^4 + 5x^3 - 7x + 9)\). So the expression is \((6x^6 - 12x^4 + 5x^3 - 7x + 9)-(8x^6 + 17x^4 - 20x^3 - 27x^2 - 6)\).

Step2: Distribute the negative sign

\(6x^6 - 12x^4 + 5x^3 - 7x + 9 - 8x^6 - 17x^4 + 20x^3 + 27x^2 + 6\).

Step3: Combine like terms

• For \(x^6\) terms: \(6x^6-8x^6=-2x^6\)
• For \(x^4\) terms: \(-12x^4-17x^4=-29x^4\)
• For \(x^3\) terms: \(5x^3 + 20x^3=25x^3\)
• For \(x^2\) terms: \(27x^2\) (since there is no other \(x^2\) term)
• For \(x\) terms: \(-7x\) (since there is no other \(x\) term)
• For constant terms: \(9 + 6=15\)

So the result is \(-2x^6 - 29x^4 + 25x^3 + 27x^2 - 7x + 15\).

Answer:

• Problem 2: \(x^5 + 15x^2 - 49\)
• Problem 3: \(-2x^6 - 29x^4 + 25x^3 + 27x^2 - 7x + 15\)