Subtract these polynomials.
$\left(2 x^2+2 x+3\right)-\left(x^2+2 x+1\right)=$
A. $x^2+4 x+2$
B. $x^2+4 x+4$
C. $x^2+4$
D. $x^2+2$
Answer (1)
Distribute the negative sign: ( 2 x 2 + 2 x + 3 ) − ( x 2 + 2 x + 1 ) = 2 x 2 + 2 x + 3 − x 2 − 2 x − 1 .
Combine like terms: ( 2 x 2 − x 2 ) + ( 2 x − 2 x ) + ( 3 − 1 ) .
Simplify: x 2 + 2 .
The final answer is x 2 + 2 .
Explanation
Understanding the Problem We are asked to subtract the polynomial ( x 2 + 2 x + 1 ) from the polynomial ( 2 x 2 + 2 x + 3 ) . This involves distributing the negative sign and combining like terms.
Distributing the Negative Sign First, distribute the negative sign to each term in the second polynomial: ( 2 x 2 + 2 x + 3 ) − ( x 2 + 2 x + 1 ) = 2 x 2 + 2 x + 3 − x 2 − 2 x − 1
Combining Like Terms Next, combine like terms (terms with the same power of x): ( 2 x 2 − x 2 ) + ( 2 x − 2 x ) + ( 3 − 1 ) = ( 2 − 1 ) x 2 + ( 2 − 2 ) x + ( 3 − 1 )
Simplifying the Polynomial Simplify the resulting polynomial: 1 x 2 + 0 x + 2 = x 2 + 2
Selecting the Correct Answer Compare the simplified polynomial with the given options. The correct answer is x 2 + 2 , which corresponds to option D.
Examples
Polynomial subtraction is used in various fields, such as engineering and computer graphics. For example, in computer graphics, subtracting polynomials can help determine the difference between two curves or surfaces, which is useful for rendering and modeling objects. In real life, if you have a budget represented by one polynomial and expenses represented by another, subtracting the expense polynomial from the budget polynomial gives you the remaining funds.
