Questions: Perform this subtraction: (9 x^5 - x^2 + 4 x) - (-3 x^5 + x^2 - 1)

Perform this subtraction: (9 x^5 - x^2 + 4 x) - (-3 x^5 + x^2 - 1)
Transcript text: Perform this subtraction: \[ \left(9 x^{5}-x^{2}+4 x\right)-\left(-3 x^{5}+x^{2}-1\right) \]
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Solution

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Solution Steps

To perform the subtraction of two polynomials, distribute the negative sign across the second polynomial and then combine like terms. This involves adding or subtracting the coefficients of terms with the same degree.

Step 1: Distribute the Negative Sign

To subtract the polynomial \(-3x^5 + x^2 - 1\) from \(9x^5 - x^2 + 4x\), first distribute the negative sign across the second polynomial: \[ -( -3x^5 + x^2 - 1 ) = 3x^5 - x^2 + 1 \]

Step 2: Combine Like Terms

Add the resulting polynomial from Step 1 to the first polynomial: \[ (9x^5 - x^2 + 4x) + (3x^5 - x^2 + 1) \]

Combine the like terms:

  • For \(x^5\) terms: \(9x^5 + 3x^5 = 12x^5\)
  • For \(x^2\) terms: \(-x^2 - x^2 = -2x^2\)
  • For \(x\) terms: \(4x\) (no like term to combine)
  • Constant term: \(1\)
Step 3: Write the Resulting Polynomial

The resulting polynomial after combining like terms is: \[ 12x^5 - 2x^2 + 4x + 1 \]

Final Answer

\[ \boxed{12x^5 - 2x^2 + 4x + 1} \]

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