Question

Add, subtract, or multiply, as indicated. Express your answer as a single polynomial in standard form. $$ \left(6 x^{5}+x^{3}+x\right)+\left(5 x^{4}-x^{3}+3 x^{2}\right) $$

Short answer

6x^5 + 5x^4 + 3x^2 + x

Step-by-step solution

- Identify and align like terms

Identify the like terms in each polynomial. The given polynomials are \ \ \( (6x^5 + x^3 + x) \) and \( (5x^4 - x^3 + 3x^2) \).

- Group the like terms

Group the like terms together. This gives \ \ \( 6x^5 \) \ \( + 5x^4 \) \ \( + x^3 - x^3 \) \ \( + 3x^2 \) \ \( + x \).

- Simplify the grouped terms

Simplify each group of like terms by combining them. This results in: \ \ \( 6x^5 + 5x^4 + (x^3 - x^3) + 3x^2 + x \) \ \ which simplifies to: \ \ \( 6x^5 + 5x^4 + 3x^2 + x \).

- Write the final polynomial in standard form

Ensure the polynomial is arranged in order of descending powers of \( x \). The final polynomial is: \ \ \( 6x^5 + 5x^4 + 3x^2 + x \).

Key concepts

Adding Polynomials

Adding polynomials might seem tricky at first, but it's actually quite straightforward once you get the hang of it. When you add polynomials, you're essentially combining two or more expressions made up of multiple terms. Each term consists of a coefficient (a number) and a variable (like x) raised to an exponent. Here's a step-by-step guide:

First, write down the polynomials you need to add. For instance, if we have \(6x^5 + x^3 + x\) and \(5x^4 - x^3 + 3x^2\), the goal is to combine them into one expression.

Next, align the like terms. Like terms are terms that have the same variables raised to the same power. It's helpful to rewrite the polynomials so that like terms are grouped together.

After grouping the like terms, proceed to combine them. This means adding their coefficients together while keeping the variable and the exponent the same. For example, in our case, \(x^3 + (-x^3)\) simplifies to \(0x^3\). By combining every pair of like terms, you get a new polynomial that represents the sum of the original expressions.

Combining Like Terms

Combining like terms is a fundamental operation in simplifying polynomials. Like terms are terms that have identical variables raised to the same power. For example, \(2x^2\) and \(5x^2\) are like terms, but \(2x^2\) and \(2x\) are not.

To combine them, simply add or subtract the coefficients while keeping the variable part unchanged. For instance, combining \(x^3 \) and \(-x^3 \) results in \(0x^3 \), effectively canceling each other out. Here's how you do it in practice:

• Identify the like terms in your polynomials. In our example, we had \(6x^5 + x^3 + x \) and \( 5x^4 - x^3 + 3x^2 \).
• Rewrite these terms by grouping them together, so they are easy to combine: \( 6x^5 + 5x^4 + (x^3 - x^3) + 3x^2 + x \).
• Combine the coefficients of the like terms: \( x^3 - x^3 \) results in \( 0 \), leaving us with the simplified terms: \( 6x^5 + 5x^4 + 3x^2 + x \).

Combining like terms helps streamline your polynomial, making it easier to work with and understand.

Standard Form of a Polynomial

The standard form of a polynomial is a specific way of arranging the terms. In standard form, the polynomial is written with its terms in descending order of their exponents. This means the term with the highest power of the variable goes first, followed by the next highest, and so on.

For example, the polynomial \(6x^5 + 5x^4 + 3x^2 + x\) is in standard form because the exponents of x decrease from 5 to 1. Here’s how to arrange terms into standard form:

First, write down all the terms of the polynomial.

• If necessary, combine like terms first to simplify the polynomial, as discussed earlier.
• Order the terms so that the one with the highest exponent comes first. Continue arranging in descending order.
• Make sure all exponents are in decreasing order: like \(ax^n + bx^{n-1} + ... + k\).

In our example, we began with \(6x^5 + x^3 + x\) and \(5x^4 - x^3 + 3x^2\). After combining like terms, we arranged them in standard form to get \(6x^5 + 5x^4 + 3x^2 + x\).

Having polynomials in standard form makes it easier to identify their degree and perform further operations like addition or subtraction.