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Find:
[tex]\left(6 m^5+3-m^3-4 m\right)-\left(-m^5+2 m^3-4 m+6\right)[/tex]

1. Write subtraction of a polynomial expression as addition of the additive inverse:
[tex]\left(6 m^5+3-m^3-4 m\right)+\left(m^5-2 m^3+4 m-6\right)[/tex]

2. Rewrite terms that are subtracted as addition of the opposite:
[tex]6 m^5+3+\left(-m^3\right)+(-4 m)+m^5+\left(-2 m^3\right)+4 m+(-6)[/tex]

3. Group like terms:
[tex]\left[6 m^5+m^5\right]+[3+(-6)]+\left[\left(-m^3\right)+\left(-2 m^3\right)\right]+[(-4 m)+4 m][/tex]

4. Combine like terms.

5. Write the resulting polynomial in standard form:
[tex]\square \ m^5 \ \square \ m^3 \ +\ \square \ m \ -\ 3[/tex]



Answer :

GinnyAnswer
Let's solve the given problem step-by-step.

Given:
[tex]\[ \left(6 m^5+3-m^3-4 m\right)-\left(-m^5+2 m^3-4 m+6\right) \][/tex]

1. Write the subtraction of the polynomial expression as the addition of the additive inverse:
[tex]\[ \left(6 m^5 + 3 - m^3 - 4 m\right) + \left(m^5 - 2 m^3 + 4 m - 6\right) \][/tex]

2. Rewrite terms that are subtracted as the addition of the opposite:
[tex]\[ 6 m^5 + 3 + (-m^3) + (-4 m) + m^5 + (-2 m^3) + 4 m + (-6) \][/tex]

3. Group like terms:
[tex]\[ \Big[6 m^5 + m^5\Big] + \Big[3 + (-6)\Big] + \Big[(-m^3) + (-2 m^3)\Big] + \Big[(-4 m) + 4 m\Big] \][/tex]

4. Combine like terms:
[tex]\[ (6 m^5 + m^5) + (3 - 6) + (-m^3 - 2 m^3) + (-4 m + 4 m) \][/tex]

Calculate each group:
[tex]\[ (6 m^5 + m^5) = 7 m^5 \][/tex]
[tex]\[ (3 - 6) = -3 \][/tex]
[tex]\[ (-m^3 - 2 m^3) = -3 m^3 \][/tex]
[tex]\[ (-4 m + 4 m) = 0 \][/tex]

5. Write the resulting polynomial in standard form:
[tex]\[ 7 m^5 - 3 m^3 + 0 m - 3 \][/tex]

Thus:
[tex]\[ \boxed{7 m^5 - 3 m^3 + 0 m - 3} \][/tex]
Which simplifies to:
[tex]\[ 7 m^5 - 3 m^3 - 3 \][/tex]

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