Subtract $\left(2 x^2-2\right)$ from $\left(6 x^2+2 x-3\right)$.
Answer (1)
Distribute the negative sign: ( 6 x 2 + 2 x − 3 ) − ( 2 x 2 − 2 ) = 6 x 2 + 2 x − 3 − 2 x 2 + 2 .
Combine like terms: ( 6 x 2 − 2 x 2 ) + 2 x + ( − 3 + 2 ) .
Simplify: 4 x 2 + 2 x − 1 .
The result of subtracting ( 2 x 2 − 2 ) from ( 6 x 2 + 2 x − 3 ) is 4 x 2 + 2 x − 1 .
Explanation
Understanding the Problem We are asked to subtract the polynomial ( 2 x 2 − 2 ) from the polynomial ( 6 x 2 + 2 x − 3 ) . This means we need to perform the operation ( 6 x 2 + 2 x − 3 ) − ( 2 x 2 − 2 ) .
Distributing the Negative Sign To subtract the polynomials, we distribute the negative sign to each term in the second polynomial:
( 6 x 2 + 2 x − 3 ) − ( 2 x 2 − 2 ) = 6 x 2 + 2 x − 3 − 2 x 2 + 2
Combining Like Terms Now, we combine like terms. We group the x 2 terms, the x terms, and the constant terms:
( 6 x 2 − 2 x 2 ) + 2 x + ( − 3 + 2 )
Simplifying the Expression Finally, we simplify each group of terms:
6 x 2 − 2 x 2 = 4 x 2 2 x remains as 2 x − 3 + 2 = − 1
So, the resulting polynomial is 4 x 2 + 2 x − 1 .
Final Answer Therefore, subtracting ( 2 x 2 − 2 ) from ( 6 x 2 + 2 x − 3 ) gives us 4 x 2 + 2 x − 1 .
Examples
Polynomial subtraction is used in various fields, such as engineering and computer graphics. For example, when designing a bridge, engineers might use polynomial subtraction to calculate the difference in stress at different points. In computer graphics, it can be used to determine the difference in color gradients or to manipulate curves and surfaces. Understanding polynomial subtraction helps in modeling and solving real-world problems in these fields.
