Question

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

Short answer

7x^2 - 3x - 11

Step-by-step solution

- Distribute the constant

Distribute the constant 3 to each term inside the parentheses: 3(2x^2 + x - 5) = 6x^2 + 3x - 15.

- Rewrite the expression

Rewrite the original expression including the distributed terms from Step 1: (x^2 - 6x + 4) + (6x^2 + 3x - 15).

- Combine like terms

Combine the like terms from the two polynomials: x^2 + 6x^2 = 7x^2, -6x + 3x = -3x, 4 - 15 = -11. The resulting polynomial is 7x^2 - 3x - 11.

- Write in standard form

Ensure that the polynomial is written in standard form, which means arranging the terms in descending order of the exponents of x: 7x^2 - 3x - 11 is already in standard form.

Key concepts

Adding Polynomials

Adding polynomials involves summing up the like terms from each polynomial.
For example, given two polynomials, such as \(x^2 - 6x + 4\) and \(6x^2 + 3x - 15\), we add the terms that have the same variables raised to the same power.
This means adding the coefficients of the \(x^2\) terms, the \(-6x\) and \(3x\) terms, and the constant terms, 4 and -15.
Once the coefficients are added, you combine them into a single polynomial.

Distributing Constants

Distributing a constant through a polynomial means multiplying each term inside the parentheses by that constant.
For example, if you have \(3(2x^2 + x - 5)\), you multiply each term within the parentheses by 3.
This results in \(3 \times 2x^2 = 6x^2\), \(3 \times x = 3x\), and \(3 \times -5 = -15\).
So, \(3(2x^2 + x - 5)\) becomes \(6x^2 + 3x - 15\).
This step is crucial before you can combine the terms with another polynomial.

Combining Like Terms

Combining like terms is the process of adding or subtracting terms with the same variable and exponent.
In the solution, you first rewrite the given expression after distributing: \( (x^2 - 6x + 4) + (6x^2 + 3x - 15) \).
Next, identify and group the like terms:

• \(x^2\) terms: \(x^2\) and \(6x^2\)
• \(-6x\) and \(3x\)
• \(4\) and \(-15\)

Then, combine them: \(x^2 + 6x^2 = 7x^2\).
\(-6x + 3x = -3x\), and \(4 - 15 = -11 \).
The final polynomial after combining like terms is \(7x^2 - 3x - 11\).

Standard Form

Expressing a polynomial in standard form means writing the terms in descending order of their exponents.
For example, \(7x^2 - 3x - 11\) is in standard form because the term with the highest exponent (\(7x^2\)) is first, followed by the next highest (\(-3x\)), and finally the constant (\(-11\)).
This format is important for consistency and ease of understanding.
If your polynomial isn't in standard form, rearrange the terms starting with the highest exponent.
Always double-check your work to ensure the terms are correctly ordered.