Question

Add, subtract, or multiply, as indicated. Express your answer as a single polynomial in standard form. $$ \left(x^{2}+6 x+8\right)+\left(3 x^{2}-4 x+7\right) $$

Short answer

4x^2 + 2x + 15

Step-by-step solution

- Identify the polynomials

Recognize the polynomials involved in the addition operation: The first polynomial is \(x^2 + 6x + 8\). The second polynomial is \(3x^2 - 4x + 7\).

- Arrange like terms

Rewrite the expression by aligning the like terms:\( (x^2 + 3x^2) + (6x - 4x) + (8 + 7) \)

- Add like terms

Add the coefficients of the like terms:\( x^2 + 3x^2 = 4x^2 \),\( 6x - 4x = 2x \),\( 8 + 7 = 15 \).So the resulting polynomial is \( 4x^2 + 2x + 15 \).

- Write the answer in standard form

The result of adding the polynomials is \( 4x^2 + 2x + 15 \). Ensure the polynomial is in standard form: the terms should be written in descending order of their exponents.

Key concepts

Adding Polynomials

When we are adding polynomials, we combine two or more polynomials to get a single polynomial. To do this, follow these steps:
1. Identify the polynomials that you need to add. Make sure you understand what each polynomial represents.
2. Align the polynomials in such a way that like terms are vertically aligned. This makes the addition process simpler.
3. Combine the like terms by adding their coefficients. Remember, only the coefficients (the numbers in front of the variables) are added, not the variables themselves.
For example, let's add the polynomials \(x^2 + 6x + 8\) and \(3x^2 - 4x + 7\).
First, we align the polynomials:

• \(x^2 + 3x^2 \)
• \(6x - 4x \)
• \(8 + 7 \)

Then, add the like terms together to get \(4x^2 + 2x + 15\).
This way, you have successfully added the polynomials.

Standard Form

The standard form of a polynomial is when its terms are ordered from highest to lowest exponent. This form makes it easier to compare and work with polynomials.
A polynomial in standard form looks like this: \(ax^n + bx^{n-1} + cx^{n-2} + \text{...} + zx^0\). Here, the exponents decrease from left to right, starting from the largest exponent.

For the polynomial \(4x^2 + 2x + 15\) that we derived by adding \(x^2 + 6x + 8\) and \(3x^2 - 4x + 7\),
It is already in standard form because the terms are ordered from highest exponent (\(x^2\)) to lowest exponent (constant term).

Like Terms

Like terms are terms that have the same variable raised to the same power. When adding polynomials, we can only combine like terms.
For instance, \(4x^2\) and \(3x^2\) are like terms because they both have \(x^2\). However, \(4x^2\) and \(2x\) are not like terms because their variables are raised to different powers.

When adding polynomials, you align like terms together:
For \(x^2 + 6x + 8\) and \(3x^2 - 4x + 7\), the like terms are:

• \(x^2\) and \(3x^2\)
• \(6x\) and \(-4x\)
• \(8\) and \(7\)

Then, you add the coefficients of the like terms: \(x^2 + 3x^2 = 4x^2\), \(6x - 4x = 2x\), and \(8 + 7 = 15\). This gives us the polynomial \(4x^2 + 2x + 15\).