Perform the indicated operations on the following polynomials.
Subtract $6 a^3-2 a^2+4$ from $5 a^3-4 a+7$
$\square$ a $\square$ + $\square$
$\square$
Answer (2)
Distribute the negative sign: ( 5 a 3 − 4 a + 7 ) − ( 6 a 3 − 2 a 2 + 4 ) = 5 a 3 − 4 a + 7 − 6 a 3 + 2 a 2 − 4 .
Combine like terms: ( 5 a 3 − 6 a 3 ) + ( 2 a 2 ) + ( − 4 a ) + ( 7 − 4 ) .
Simplify: − a 3 + 2 a 2 − 4 a + 3 .
The result of the subtraction is − a 3 + 2 a 2 − 4 a + 3 .
Explanation
Understanding the problem We are asked to subtract the polynomial 6 a 3 − 2 a 2 + 4 from the polynomial 5 a 3 − 4 a + 7 . This means we need to compute ( 5 a 3 − 4 a + 7 ) − ( 6 a 3 − 2 a 2 + 4 ) .
Distributing the negative sign First, distribute the negative sign to each term in the second polynomial: 5 a 3 − 4 a + 7 − 6 a 3 + 2 a 2 − 4 .
Combining like terms Next, we combine like terms. We group the terms with the same power of a together: ( 5 a 3 − 6 a 3 ) + ( 2 a 2 ) + ( − 4 a ) + ( 7 − 4 ) .
Simplifying the expression Now, we simplify each group of like terms:
5 a 3 − 6 a 3 = − 1 a 3 = − a 3 2 a 2 = 2 a 2 − 4 a = − 4 a 7 − 4 = 3
So the resulting polynomial is − a 3 + 2 a 2 − 4 a + 3 .
Final Answer Therefore, the result of subtracting 6 a 3 − 2 a 2 + 4 from 5 a 3 − 4 a + 7 is − a 3 + 2 a 2 − 4 a + 3 .
Examples
Polynomial subtraction is used in various applications, such as calculating the change in revenue or cost over time. For example, if you have two different investment plans, you can use polynomial subtraction to determine the difference in their projected growth. Understanding polynomial operations helps in making informed financial decisions.
To subtract the polynomials, we distribute the negative sign and combine like terms. The final result is − a 3 + 2 a 2 − 4 a + 3 . This demonstrates polynomial subtraction through clear steps of distribution and combination of terms.
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