Question

Multiply the polynomials using the special product formulas. Express your answer as a single polynomial in standard form. $$ (2 x+1)^{3} $$

Short answer

The polynomial is \ 8x^3 + 12x^2 + 6x + 1.

Step-by-step solution

- Identify the special product formula

Recognize that \( (2x + 1)^3 \) is a cubic binomial. The special product formula for a binomial raised to the third power is \[ (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3. \] In this problem, \( a = 2x \) and \( b = 1. \)

- Substitute into the formula

Substitute \( a = 2x \) and \( b = 1 \) into the formula: \[(2x + 1)^3 = (2x)^3 + 3(2x)^2(1) + 3(2x)(1^2) + 1^3. \]

- Simplify each term

Calculate each term separately: \[(2x)^3 = 8x^3 \] \[ 3(2x)^2(1) = 3(4x^2)(1) = 12x^2 \] \[ 3(2x)(1^2) = 3(2x)(1) = 6x \] \[ 1^3 = 1. \]

- Combine the terms

Combine the simplified terms to form the polynomial: \[(2x + 1)^3 = 8x^3 + 12x^2 + 6x + 1. \]

Key concepts

Special Product Formulas

Special product formulas are a set of algebraic rules that simplify the multiplication of polynomials. One useful special product formula is the binomial expansion of \(a + b\) raised to a power. For instance, the binomial theorem for cubes (\

Cubic Binomial

A cubic binomial is any binomial expression raised to the third power. In this exercise, \(2x + 1\) is the cubic binomial. Cubic binomials can be expanded using the special product formula for cubes. By identifying \(a\) and \(b\) in \(a + b\) early on, we can apply the formula efficiently. In our specific example, \(a = 2x\) and \(b = 1\). Here is a quick breakdown of how this works:

• \((2x)^3 = 8x^3\)
• \(3(2x)^2(1) = 12x^2\)
• \(3(2x)(1^2) = 6x\)
• \(1^3 = 1\)

Hence, the product of a cubic binomial is obtained by summing up these simplified terms.

Polynomial Standard Form

The standard form of a polynomial is when its terms are presented in descending order of their degrees. For our expanded cubic binomial, the polynomial in standard form is: \(8x^3 + 12x^2 + 6x + 1\). Breaking it down:

• \ 8x^3: Cubic term\
• \ 12x^2: Quadratic term\
• \ 6x: Linear term\
• \ 1: Constant term\

Presenting a polynomial in standard form makes it easier to read and work with, especially when performing additional algebraic operations such as addition, subtraction, or differentiation.

Binomial Expansion

Binomial expansion involves expressing a binomial raised to any power (like 2, 3, 4, etc.) as a sum of terms. The binomial theorem provides a systematic way to expand binomials. For example, \( (a + b)^n \) can be expanded using combinations and powers of \(a\) and \b. The formula for the expansion of a cubic binomial follows: \((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\). This helps in converting exponential expressions into polynomial standard form. For the given exercise: \ ((2x + 1)^3\}, we used this principle to successfully expand and combine the terms into a single polynomial.