$\left(4.5 x^4+3 x^3-x^2+6\right)+\left(2.6 x^4-3 x+9\right)$

Question

kimberly651apple kimberly651apple

03-11-2026
Mathematics

Answered

$\left(4.5 x^4+3 x^3-x^2+6\right)+\left(2.6 x^4-3 x+9\right)$

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-03T13:01:09-05:00

Group the terms of the two polynomials by their exponents: $x^4, x^3, x^2, x^1,$ and constants.
Add the coefficients for each corresponding power of $x$.
Combine the results: $(4.5 + 2.6)x^4 = 7.1x^4$, $3x^3 = 3x^3$, $-x^2 = -x^2$, $-3x = -3x$, and $6 + 9 = 15$.
The simplified polynomial is $\boxed{7.1x^4 + 3x^3 - x^2 - 3x + 15}$.

Explanation

Analysis of the Polynomials

To simplify the expression $(4.5 x^4+3 x^3-x^2+6)+(2.6 x^4-3 x+9)$, we identify and combine the terms that have the same power of $x$. The given polynomials are $P(x) = 4.5 x^4+3 x^3-x^2+6$ and $Q(x) = 2.6 x^4-3 x+9$. We are asked to compute their sum, $P(x) + Q(x)$.

Grouping and Combining Like Terms

We rewrite the expression by grouping terms with the same degree (power of $x$):
\[(4.5 x^4 + 2.6 x^4) + 3 x^3 - x^2 - 3 x + (6 + 9)\]
Now, we perform the arithmetic for each group:

For $x^4$: $4.5 + 2.6 = 7.1$
For $x^3$: $3$
For $x^2$: $-1$
For $x^1$: $-3$
For the constant terms: $6 + 9 = 15$

Combining these, we get:
\[7.1 x^4 + 3 x^3 - x^2 - 3 x + 15\]
This is the simplified polynomial resulting from the addition.

Conclusion

The simplified sum of the two polynomials is $7.1x^4 + 3x^3 - x^2 - 3x + 15$. We have combined the coefficients of corresponding powers of $x$ correctly to arrive at the final expression. Thus, the final answer is $\boxed{7.1x^4 + 3x^3 - x^2 - 3x + 15}$.

Examples

In physics and engineering, polynomial addition is often used to calculate total quantities. For example, if two different forces acting on an object are represented as polynomials depending on the object's displacement $x$, their combined effect is found by adding these polynomials. This allows scientists to model complex systems by simplifying combined interactions into a single manageable equation.