Find the sum and express it in simplest form.
$\left(-5 x^3+6 x-4\right)+\left(-2 x^3+x\right)$
Answer (1)
Combine like terms by grouping terms with the same power of x .
Add the coefficients of the like terms: ( − 5 x 3 − 2 x 3 ) + ( 6 x + x ) − 4 = ( − 5 − 2 ) x 3 + ( 6 + 1 ) x − 4 .
Simplify the expression: − 7 x 3 + 7 x − 4 .
The sum of the polynomials in simplest form is − 7 x 3 + 7 x − 4 .
Explanation
Understanding the Problem We are given two polynomials: ( − 5 x 3 + 6 x − 4 ) and ( − 2 x 3 + x ) . Our goal is to find their sum and express the result in simplest form. This involves combining like terms, which are terms with the same power of x .
Combining Like Terms To add the two polynomials, we combine like terms: ( − 5 x 3 + 6 x − 4 ) + ( − 2 x 3 + x ) We group the terms with the same power of x :
( − 5 x 3 − 2 x 3 ) + ( 6 x + x ) − 4 Now, we combine the coefficients of the like terms: ( − 5 − 2 ) x 3 + ( 6 + 1 ) x − 4
Simplifying the Expression Now we simplify the expression by performing the arithmetic: − 7 x 3 + 7 x − 4 The simplified sum is − 7 x 3 + 7 x − 4 .
Final Answer Therefore, the sum of the given polynomials in simplest form is − 7 x 3 + 7 x − 4 .
Examples
Polynomials are used in various fields like physics, engineering, and economics. For example, in physics, the trajectory of a projectile can be modeled using a polynomial function. In economics, cost and revenue functions are often expressed as polynomials. Understanding how to add and simplify polynomials is essential for solving problems in these areas.
