Question

Add, subtract, or multiply, as indicated. Express your answer as a single polynomial in standard form. $$ \left(x^{2}+1\right)-\left(4 x^{2}+5\right)+\left(x^{2}+x-2\right) $$

Short answer

-2x^2 + x - 6

Step-by-step solution

- Distribute subtraction

First, distribute the subtraction sign through the second polynomial: \( -(4x^2 + 5) \). This results in: \(-4x^2 - 5\).

- Combine all polynomials

Rewrite the expression combining all the polynomials: \( (x^2 + 1) + (-4x^2 - 5) + (x^2 + x - 2) \).

- Group like terms

Combine like terms by grouping them together:\[ (x^2 - 4x^2 + x^2) + (x) + (1 - 5 - 2) \].

- Simplify each group

Simplify each group of like terms:\[ (-2x^2) + x + (-6) \].

- Write in standard form

Combine the simplified groups into a single polynomial: \(-2x^2 + x - 6\). This is the expression in standard form.

Key concepts

polynomial subtraction

Polynomial subtraction involves distributing the subtraction sign through the polynomial you are subtracting and then combining like terms. Remember that subtracting a polynomial is the same as adding its opposite. For example, in the original exercise, we subtract \(4x^2 + 5\) by distributing the negative sign to each term inside the parentheses. This gives us \(-4x^2 - 5\).
Here are some tips to make polynomial subtraction easier:

• Always distribute the minus sign to each term inside the parentheses.
• Be careful with your signs; a common mistake is forgetting to change the signs of every term.
• It may help to write down the opposite polynomial separately before combining.
• After distributing, combine like terms to simplify the expression.

polynomial addition

Polynomial addition is a fundamental operation where we combine polynomials by adding the coefficients of like terms. In the provided exercise, after distributing the negative sign, we have three polynomials to combine: \(x^2 + 1\), \(-4x^2 - 5\), and \(x^2 + x - 2\).
Here's how to do it step by step:

• Align the like terms vertically or mentally group them.
• Add the coefficients of like terms together.
• In our exercise, combine the \(x^2\) terms: \(x^2 - 4x^2 + x^2 = -2x^2\).
• Then, combine the constant terms: \(1 - 5 - 2 = -6\).
• Don't forget to add the \(x\) term separately, which remains \(+x\).

This operation will result in the simplified polynomial \(-2x^2 + x - 6\).

standard form

The standard form of a polynomial arranges the terms in order of descending powers of the variable. For example, a polynomial in the form \ax^n + bx^{n-1} + ... + k\, where a, b, ..., k are constants and n is a non-negative integer, is considered to be in standard form.
In our exercise, after performing addition and subtraction of the given polynomials, we get \(-2x^2 + x - 6\).
Notice:

• The highest degree term, \-2x^2\, comes first.
• Followed by the linear term, \+x\.
• Lastly, the constant term, \-6\.

Writing polynomials in standard form is important because it provides a consistent way of presenting and working with polynomials. It also makes comparison and further operations more straightforward.