Question

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

Short answer

The polynomial in standard form is \(x^3 + 3x^2 - 2x - 4\).

Step-by-step solution

Distribute x

Multiply each term in the second polynomial by the first term in the first polynomial, which is \(x\). \[ x \times \big(x^2 + 2x - 4\big) = x^3 + 2x^2 - 4x \]

Distribute 1

Multiply each term in the second polynomial by the second term in the first polynomial, which is \(1\). \[ 1 \times \big(x^2 + 2x - 4\big) = x^2 + 2x - 4 \]

Combine like terms

Add the results from Step 1 and Step 2 together and arrange terms in descending order of exponents.\[ x^3 + 2x^2 - 4x + x^2 + 2x - 4 \]Combine the like terms:\[ x^3 + (2x^2 + x^2) + (-4x + 2x) - 4 = x^3 + 3x^2 - 2x - 4 \]

Key concepts

Distributive Property

The distributive property is a key concept in algebra. It lets you multiply a single term by each term within a set of parentheses. For example, in the expression ewline (x+1)(x^2 + 2x - 4),ewline you distribute each term in the first polynomial across the terms in the second polynomial.
This means multiplying each term individually: ewline x(x^2 + 2x - 4)ewline and then 1(x^2 + 2x - 4).The distributive property helps to expand polynomials and simplify expressions. This step-by-step process ensures you account for every term correctly.

Combining Like Terms

Combining like terms means gathering all similar terms together. These are terms that have the same variable raised to the same power. For example, in our expression, after distributing, we have: ewline x^3 + 2x^2 - 4x + x^2 + 2x - 4.
To combine like terms:

• Identify the terms with the same power of x. Here, you have two x^2 terms: 2x^2 and x^2.
• Add their coefficients: 2x^2 + x^2 = 3x^2.
• Do the same for the x terms: -4x + 2x = -2x.
• Keep the constant term as is: -4.After combining, the polynomial becomes: ewline x^3 + 3x^2 - 2x - 4.
  

Polynomial Multiplication

Polynomial multiplication involves distributing each term in the first polynomial to every term in the second polynomial. Let's break down the multiplication in our example:
Firstly, distribute x to each term in (x^2 + 2x - 4):

• x * x^2 = x^3.
• x * 2x = 2x^2.
• x * (-4) = -4x. Secondly, distribute 1 to each term in (x^2 + 2x - 4):
  • 1 * x^2 = x^2.
  • 1 * 2x = 2x.
  • 1 * (-4) = -4.
  Combining these results:x^3 + 2x^2 - 4x + x^2 + 2x - 4.
  This step ensures all parts of the polynomials are multiplied.

Standard Form of a Polynomial

The standard form of a polynomial arranges the terms in descending order by the degree of each term. After multiplying and combining like terms, we write our result in standard form. For our example:We have: x^3 + 3x^2 - 2x - 4.

• The term with the highest degree (3) is x^3.
• The next highest degree (2) is 3x^2.
• Then comes -2x (degree 1).
• The constant term is -4 (degree 0).

Always express your final answer in this order. It helps in comparing polynomials and understanding their structure. Standard form makes polynomials clearer and more consistent.