Question Subtract two polynomials and simplify the result: $(-3x^5 + 9x^4 - 2x^3 + 2x^2) - (11x^5 + 7x^3 + 8x^2 + 6x)$ .

Studdy Solution

STEP 1

Assumptions
1. We are given two polynomials to subtract: $-3 x^{5}+9 x^{4}-2 x^{3}+2 x^{2}$ and $11 x^{5}+7 x^{3}+8 x^{2}+6 x$ .
2. We need to subtract the second polynomial from the first one.
3. We will combine like terms to simplify the result.

STEP 2

To subtract the second polynomial from the first, we change the sign of each term in the second polynomial and then add it to the corresponding term in the first polynomial.
$(-3 x^{5}+9 x^{4}-2 x^{3}+2 x^{2}) - (11 x^{5}+7 x^{3}+8 x^{2}+6 x)$ $= -3 x^{5}+9 x^{4}-2 x^{3}+2 x^{2} - 11 x^{5} - 7 x^{3} - 8 x^{2} - 6 x$

STEP 3

Combine like terms by adding the coefficients of the terms with the same degree of $x$ .
For $x^{5}$ terms: $-3 x^{5} - 11 x^{5} = (-3 - 11) x^{5} = -14 x^{5}$

STEP 4

For $x^{4}$ terms: Since there is no $x^{4}$ term in the second polynomial, we simply bring down the $x^{4}$ term from the first polynomial.
$9 x^{4}$

STEP 5

For $x^{3}$ terms: $-2 x^{3} - 7 x^{3} = (-2 - 7) x^{3} = -9 x^{3}$

STEP 6

For $x^{2}$ terms: $2 x^{2} - 8 x^{2} = (2 - 8) x^{2} = -6 x^{2}$

STEP 7

For $x$ terms: Since there is no $x$ term in the first polynomial, we change the sign of the $x$ term from the second polynomial.
$- 6 x$

STEP 8

Combine all the simplified terms to get the final simplified polynomial.
$-14 x^{5} + 9 x^{4} - 9 x^{3} - 6 x^{2} - 6 x$

STEP 9

Write down the final simplified result.
$\boxed{-14 x^{5} + 9 x^{4} - 9 x^{3} - 6 x^{2} - 6 x}$
This is the simplified form of the polynomial after subtraction.

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