Find the sum of the polynomials. Simplify your answer.
$(2 b-9.4)+\left(-1.4 b^2-4 b+2.3\right)$
We are asked to find the sum of two polynomials: \(P_1(b) = 2b - 9.4\) and \(P_2(b) = -1.4b^2 - 4b + 2.3\). To compute the sum, we need to combine the expressions and then simplify the result by grouping like terms. Like terms are terms that have the same power of the variable \(b\). In this case, we have a quadratic term (\(b^2\)), a linear term (\(b^1\)), and a constant term (\(b^0\)).
To find the sum, we write:
\[(2b - 9.4) + (-1.4b^2 - 4b + 2.3)\]
Next, we remove the parentheses and group the like terms together, arranging them in descending order of the exponent of \(b\):
\[-1.4b^2 + (2b - 4b) + (-9.4 + 2.3)\]
This allows us to handle each group separately.
Now we perform the arithmetic for each group:
Combining these results, the sum is:
\[-1.4b^2 - 2b - 7.1\]
This is the standard form of the resulting polynomial.
Polynomial addition is used frequently in physics and engineering to combine physical quantities that depend on the same variable. For instance, if you are modeling the position of an object in 2D space where each coordinate is influenced by different forces, you might need to add polynomial expressions describing these forces over time. By combining like terms, you can simplify complex models to better understand the overall motion of the system, such as finding the net force at any given time \(b\).