What is the sum of the polynomials?
$\left(6 x+7+x^2\right)+\left(2 x^2-3\right)$
Answer (1)
Combine like terms of the polynomials.
Add the x 2 terms: x 2 + 2 x 2 = 3 x 2 .
Add the x terms: 6 x .
Add the constant terms: 7 + ( − 3 ) = 4 . The sum is 3 x 2 + 6 x + 4 .
Explanation
Understanding the Problem Let's analyze the problem. We are asked to find the sum of two polynomials: ( 6 x + 7 + x 2 ) and ( 2 x 2 − 3 ) . To do this, we need to combine like terms.
Rewriting Polynomials First, let's rewrite the first polynomial in standard form (decreasing order of exponents): x 2 + 6 x + 7 . The second polynomial is already in a simple form: 2 x 2 − 3 .
Adding the Polynomials Now, we add the two polynomials together: ( x 2 + 6 x + 7 ) + ( 2 x 2 − 3 ) We combine the x 2 terms: x 2 + 2 x 2 = 3 x 2 .
We have only one x term: 6 x .
We combine the constant terms: 7 − 3 = 4 .
So, the sum is 3 x 2 + 6 x + 4 .
Final Answer Comparing our result with the given options, we see that the sum of the polynomials is 3 x 2 + 6 x + 4 .
Examples
Polynomials are used in many areas of mathematics and science. For example, they are used to model curves and trajectories in physics, such as the path of a ball thrown in the air. They are also used in engineering to design structures and circuits. In economics, polynomials can be used to model cost and revenue functions. Understanding how to add polynomials is a fundamental skill that can be applied in various real-world scenarios.
