Subtract: [tex]x^2+2 x+1-(x-3)[/tex]
Answer (1)
Distribute the negative sign: x 2 + 2 x + 1 − ( x − 3 ) = x 2 + 2 x + 1 − x + 3 .
Combine like terms: x 2 + ( 2 x − x ) + ( 1 + 3 ) .
Simplify the expression: x 2 + x + 4 .
The final result is x 2 + x + 4 .
Explanation
Understanding the problem We are asked to subtract the polynomial ( x − 3 ) from the polynomial ( x 2 + 2 x + 1 ) . This means we need to simplify the expression x 2 + 2 x + 1 − ( x − 3 ) .
Distributing the negative sign First, we distribute the negative sign to the terms inside the parenthesis: x 2 + 2 x + 1 − ( x − 3 ) = x 2 + 2 x + 1 − x + 3
Combining like terms Next, we combine like terms. We have the x 2 term, the x terms ( 2 x and − x ), and the constant terms ( 1 and 3 ). Combining these, we get: x 2 + ( 2 x − x ) + ( 1 + 3 ) x 2 + x + 4
Final result Therefore, the simplified expression is x 2 + x + 4 .
Examples
Polynomial subtraction is a fundamental concept in algebra and is used in various applications. For example, in physics, you might use polynomial subtraction to find the difference in the position of an object at two different times, where the position is described by a polynomial function of time. In economics, you could use it to determine the change in revenue or cost when production levels vary, with revenue and cost modeled as polynomials. Understanding polynomial subtraction helps in simplifying complex expressions and solving real-world problems involving rates of change and differences.
