Find: $(6 m^5+3-m^3-4 m)-(-m^5+2 m^3-4 m+6)$
1. Write subtraction of a polynomial expression as addition of the additive inverse.
$(6 m^5+3-m^3-4 m)+(m^5-2 m^3+4 m-6)$
2. Rewrite terms that are subtracted as addition of the opposite.
$6 m^5+3+(-m^3)+(-4 m)+m^5+(-2 m^3)+4 m+(-6)$
3. Group like terms.
$[6 m^5+m^5]+[3+(-6)]+[(-m^3)+(-2 m^3)]+[(-4 m)+4 m]$
4. Combine like terms.
5. Write the resulting polynomial in standard form.
$\square$ $m^5$ $\square$ $m^3+$ $\square$ $m-3
Answer (2)
To simplify the expression ( 6 m 5 + 3 − m 3 − 4 m ) − ( − m 5 + 2 m 3 − 4 m + 6 ) , we combine like terms after distributing the negative sign. The final simplified polynomial is 7 m 5 − 3 m 3 − 3 .
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Distribute the negative sign: ( 6 m 5 + 3 − m 3 − 4 m ) − ( − m 5 + 2 m 3 − 4 m + 6 ) = 6 m 5 + 3 − m 3 − 4 m + m 5 − 2 m 3 + 4 m − 6 .
Group like terms: ( 6 m 5 + m 5 ) + ( − m 3 − 2 m 3 ) + ( − 4 m + 4 m ) + ( 3 − 6 ) .
Combine like terms: 7 m 5 − 3 m 3 + 0 m − 3 .
Write in standard form: 7 m 5 − 3 m 3 − 3 . The coefficients are 7, -3, 0, and -3. The final answer is 7 m 5 − 3 m 3 − 3 .
Explanation
Understanding the Problem We are asked to simplify the polynomial expression ( 6 m 5 + 3 − m 3 − 4 m ) − ( − m 5 + 2 m 3 − 4 m + 6 ) . This involves combining like terms after distributing the negative sign.
Distributing the Negative Sign First, we distribute the negative sign in the second polynomial:
( 6 m 5 + 3 − m 3 − 4 m ) − ( − m 5 + 2 m 3 − 4 m + 6 ) = 6 m 5 + 3 − m 3 − 4 m + m 5 − 2 m 3 + 4 m − 6
Grouping Like Terms Next, we group like terms:
( 6 m 5 + m 5 ) + ( − m 3 − 2 m 3 ) + ( − 4 m + 4 m ) + ( 3 − 6 )
Combining Like Terms Now, we combine like terms:
( 6 + 1 ) m 5 + ( − 1 − 2 ) m 3 + ( − 4 + 4 ) m + ( 3 − 6 ) = 7 m 5 − 3 m 3 + 0 m − 3
Writing in Standard Form Finally, we write the polynomial in standard form, which means arranging the terms in descending order of the exponent of the variable. In this case, we have:
7 m 5 − 3 m 3 − 3
So, the simplified polynomial is 7 m 5 − 3 m 3 − 3 .
Final Answer Thus, the simplified expression is 7 m 5 − 3 m 3 − 3 .
Examples
Polynomials are used to model curves and shapes in engineering and computer graphics. Simplifying polynomial expressions allows engineers to optimize designs, predict behavior, and efficiently represent complex systems. For example, simplifying the equation for the trajectory of a projectile helps in predicting its landing point, which is crucial in fields like aerospace and defense.
