Question

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

Short answer

\(x^4 + 2x^3 - 5x^2\)

Step-by-step solution

- Distribute x^2

Multiply each term inside the parentheses by the term outside the parentheses, which is \(x^2\).

- Multiply x^2 by x^2

Multiply \(x^2\) by \(x^2\) to get \(x^4\).

- Multiply x^2 by 2x

Multiply \(x^2\) by \(2x\) to get \(2x^3\).

- Multiply x^2 by -5

Multiply \(x^2\) by \(-5\) to get \(-5x^2\).

- Combine all terms

Write all the resulting terms together to get the final polynomial: \(x^4 + 2x^3 - 5x^2\).

Key concepts

distributive property

The distributive property is a fundamental concept in algebra. It allows us to simplify expressions where a single term is multiplied by terms inside parentheses.

When using the distributive property, you multiply each term inside the parentheses by the term outside. For example, in the polynomial expression \(x^{2}(x^{2} + 2x - 5)\), you apply the distributive property by multiplying \(x^{2}\) with each term inside the parentheses.

This results in:

• \(x^{2} \times x^{2} = x^{4}\)
• \(x^{2} \times 2x = 2x^{3}\)
• \(x^{2} \times -5 = -5x^{2}\)

Each term is handled separately and then summed together at the end. This makes it easier to handle and combine terms in a polynomial.

multiplying polynomials

Multiplying polynomials involves applying the distributive property to each term in the polynomial. This means that every term in the first polynomial is multiplied by every term in the second polynomial.

In the current exercise, however, you are multiplying a single term \(x^{2}\) by a trinomial \(x^{2} + 2x - 5\). After distributing the \(x^{2}\), we get each term multiplied as follows:

• \(x^{2} \times x^{2} = x^{4}\)
• \(x^{2} \times 2x = 2x^{3}\)
• \(x^{2} \times -5 = -5x^{2}\)

It is important to follow each multiplication step carefully to avoid mistakes, making note of positive and negative signs.

standard form of a polynomial

The standard form of a polynomial is where all terms are written in descending order according to their degrees. A polynomial's degree is the highest exponent of its variable.

In the exercise, after multiplying and distributing terms, we get:

\(x^{4} + 2x^{3} - 5x^{2}\)

This is already in standard form because the terms are arranged from the highest degree (\(x^{4}\)) to the lowest degree (\(-5x^{2}\)).

Writing polynomials in standard form helps in easily combining like terms and simplifies solving and understanding the polynomial's structure.

combining like terms

Combining like terms means merging terms that have the same degree, or power, of the variable. In polynomials, terms with the same exponent can be added or subtracted.

Example: \(3x^{2} + 2x^{2} - x^{2} = 4x^{2}\)

In our exercise, by distributing \(x^{2}\), we get three terms: \(x^{4}\), \(2x^{3}\), and \(-5x^{2}\). Each term has a different exponent: 4, 3, and 2, respectively.

Since there are no like terms (terms with the same degree) in \(x^{4} + 2x^{3} - 5x^{2}\), they are simply written together in standard form. Always remember, combining like terms helps simplify and solve polynomials efficiently.