Question

Add, subtract, or multiply, as indicated. Express your answer as a single polynomial in standard form. $$ \left(x^{2}-x+2\right)+\left(2 x^{2}-3 x+5\right)-\left(x^{2}+1\right) $$

Short answer

The polynomial in standard form is \(2x^2 - 4x + 6\).

Step-by-step solution

- Distribute the subtraction

Rewrite the expression \((x^2 - x + 2) + (2x^2 - 3x + 5) - (x^2 + 1)\)by distributing the negative sign through the last polynomial. This gives us:\((x^2 - x + 2) + (2x^2 - 3x + 5) - x^2 - 1\)

- Combine like terms

Group and combine the like terms from the polynomials: \[ (x^2 + 2x^2 - x^2) + (-x - 3x) + (2 + 5 - 1) \]which results in: \[ 2x^2 - 4x + 6 \]

- Write the polynomial in standard form

The polynomial \(2x^2 - 4x + 6\) is already in standard form. Thus, no further steps are needed.

Key concepts

Polynomial Addition

Polynomial addition is a common operation in algebra. To add polynomials, you combine like terms. Like terms are terms that have the same variable raised to the same power.
Here's the polynomial we want to add: \ (x^2 - x + 2) + (2x^2 - 3x + 5)
To add these polynomials, we align the like terms and add their coefficients.
For example:

• \textendash The terms with \(x^2\) are \(x^2\) and \(2x^2\)
• \textendash The terms with \(x\) are \(-x\) and \(-3x\)
• \textendash The constant terms are 2 and 5

Combining the coefficients of like terms, we get:

• \ (1 + 2)x^2 = 3x^2
• \ (-1 - 3)x = -4x
• \ (2 + 5) = 7

Therefore, the sum is \(3x^2 - 4x + 7\).
This method of adding coefficients and combining like terms can simplify and solve polynomial addition problems more efficiently.

Polynomial Subtraction

Polynomial subtraction is similar to addition but involves distributing the negative sign before combining like terms.
Let's use our original expression:
\( (x^2 - x + 2) + (2x^2 - 3x + 5) - (x^2 + 1) \)
First, we distribute the negative sign through the last polynomial, changing its signs:
\( (x^2 - x + 2) + (2x^2 - 3x + 5) - x^2 - 1 \)
Now, we combine like terms. Breaking it down:

• Terms with \(x^2\): \(x^2 + 2x^2 - x^2\)
• Terms with \(x\): \( -x - 3x \)
• Constant terms: \(2 + 5 - 1\)

Combining these, we get:
\( (1 + 2 -1)x^2 = 2x^2\)
\( (-1 - 3)x = -4x\)
\( (2 + 5 - 1) = 6\)
Thus, the result of the subtraction is \(2x^2 - 4x + 6\).
The key is to distribute negative signs correctly and then systematically combine like terms.

Standard Form of Polynomials

The standard form of a polynomial arranges its terms in descending order of the degrees of the variables.
The general structure is: \(ax^n + bx^{n-1} + \textellipsis + k\)
where \(a, b, \textellipsis ,k\) are coefficients and \(n\) is the degree of the polynomial.
In the given problem, the resulting polynomial was: \(2x^2 - 4x + 6\)
To ensure it is in the standard form, we check the order of terms:

• The term with the highest degree is \(2x^2\)
• Next is the term with \(-4x\)
• Finally, the constant 6

Since the terms are ordered from highest to lowest degree, the polynomial is in standard form.
Standard form helps to easily convey the structure and degree of a polynomial, making it simpler to understand and use in further calculations.