Question

Add, subtract, or multiply, as indicated. Express your answer as a single polynomial in standard form. $$ \left(10 x^{5}-8 x^{2}\right)+\left(3 x^{3}-2 x^{2}+6\right) $$

Short answer

10x^{5} + 3x^{3} - 10x^{2} + 6

Step-by-step solution

- Identify the Polynomials

First, identify the polynomials within the parentheses. The given polynomials are: 1) \(10x^{5} - 8x^{2}\) 2) \(3x^{3} - 2x^{2} + 6\)

- Arrange the Polynomials for Addition

Write the polynomials in a line to facilitate their addition. Thus, it becomes: \( (10x^{5} - 8x^{2}) + (3x^{3} - 2x^{2} + 6) \)

- Group Like Terms

Rearrange the terms to place like terms together:\(10x^{5} + 3x^{3} - 8x^{2} - 2x^{2} + 6 \)

- Combine Like Terms

Add the coefficients of the like terms:\(10x^{5} + 3x^{3} + (-8x^{2} - 2x^{2}) + 6\) becomes:\(10x^{5} + 3x^{3} - 10x^{2} + 6\)

- Write the Polynomial in Standard Form

Ensure the polynomial is in descending order of exponent terms:\(10x^{5} + 3x^{3} - 10x^{2} + 6\)

Key concepts

polynomial addition

Polynomial addition involves combining two or more polynomials by adding their corresponding terms. In the given exercise, two polynomials are to be added: \(10x^{5} - 8x^{2}\) and \(3x^{3} - 2x^{2} + 6\). This method follows these simple steps:

* Write down the polynomials you want to add together.
* Align the like terms vertically to easily identify which terms can be combined. Like terms have the same variable raised to the same power.
* Add the coefficients of the like terms. The variable part remains unchanged.

Remember, addition of polynomials is simply an extension of the addition of numbers. You just add the parts that are similar and carry the result forward. Finally, ensure the terms are combined properly without missing any detail.

combining like terms

Combining like terms is a crucial step in simplifying polynomials. Like terms in a polynomial have the same variables raised to the same powers. When you have like terms, you can combine them by adding or subtracting their coefficients.

In the step from our example:
\(10x^{5} - 8x^{2}\) and \(3x^{3} - 2x^{2} + 6\), we first group the like terms:

*\(10x^{5}\)
*\(3x^{3}\)
*\(-8x^{2}\) and \(-2x^{2}\)
*\(6\)

Next, we combine them:

\(-8x^{2} - 2x^{2} = -10x^{2}\).

Hence, the resulting polynomial after combining like terms is:
\(10x^{5} + 3x^{3} - 10x^{2} + 6\).

Combining like terms makes the polynomial shorter and easier to understand.

standard form of a polynomial

The standard form of a polynomial is a way of writing the polynomial such that its terms are ordered in descending powers of its variable. This makes it easy to compare and evaluate polynomials.

In our example polynomial:
\(10x^{5} + 3x^{3} - 10x^{2} + 6\),

The terms are already arranged generally from the highest power term \(10x^{5}\) to the constant term \(6\).

When putting a polynomial in standard form, remember:

* Start with the highest degree term (highest exponent on the variable).
* Continue descending until you reach the constant term (which is the term without the variable).

So, the process ensures the polynomial is organized for further operations or evaluations.