What is the sum of the given polynomials in standard form? $(x^2-3 x)+(-2 x^2+5 x-3)$
A. $-3 x^2+8 x-3$
B. $-x^2-2 x-3$
C. $3 x^2-8 x+3$
D. $-x^2+2 x-3$
Answer (2)
The sum of the polynomials ( x 2 − 3 x ) + ( − 2 x 2 + 5 x − 3 ) is found by combining like terms. After adding the x 2 terms, x terms, and constant terms, the result in standard form is − x 2 + 2 x − 3 . Therefore, the correct answer is option D.
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Combine the x 2 terms: x 2 + ( − 2 x 2 ) = − x 2 .
Combine the x terms: − 3 x + 5 x = 2 x .
Combine the constant terms: 0 + ( − 3 ) = − 3 .
Write the resulting polynomial in standard form: − x 2 + 2 x − 3 .
Explanation
Understanding the Problem We are given two polynomials: ( x 2 − 3 x ) and ( − 2 x 2 + 5 x − 3 ) . Our goal is to find their sum and express the result in standard form.
Combining Like Terms To find the sum, we combine like terms. This means we add the coefficients of the terms with the same power of x .
Combining x 2 Terms First, let's combine the x 2 terms: x 2 + ( − 2 x 2 ) = ( 1 − 2 ) x 2 = − x 2 .
Combining x Terms Next, let's combine the x terms: − 3 x + 5 x = ( − 3 + 5 ) x = 2 x .
Combining Constant Terms Finally, let's combine the constant terms: 0 + ( − 3 ) = − 3 .
Writing in Standard Form Now, we write the resulting polynomial in standard form, which means ordering the terms by decreasing degree: − x 2 + 2 x − 3 .
Final Answer Therefore, the sum of the given polynomials in standard form is − x 2 + 2 x − 3 .
Examples
Polynomials are used to model various real-world phenomena. For example, the trajectory of a projectile (like a ball thrown in the air) can be described by a quadratic polynomial. Adding polynomials can help combine different models or analyze the combined effect of multiple factors. In engineering, polynomials are used to design curves, surfaces, and optimize processes. Understanding polynomial addition is crucial for solving problems in physics, engineering, and computer graphics.
