Question

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

Short answer

4x^2 - 12x + 9

Step-by-step solution

Identify the special product formula

Recognize that the expression \( (2x - 3)^2 \) can be expanded using the special product formula for the square of a binomial, which is \( (a - b)^2 = a^2 - 2ab + b^2 \).

Assign values to the variables

In the expression \( (2x - 3)^2 \), assign \( a = 2x \) and \( b = 3 \).

Apply the formula

Substitute \( a \) and \( b \) into the formula \ (a - b)^2 = a^2 - 2ab + b^2 \[ (2x - 3)^2 = (2x)^2 - 2(2x)(3) + (3)^2 \]

Perform individual calculations

Calculate each term separately:\( (2x)^2 = 4x^2 \)\( 2(2x)(3) = 12x \)\( (3)^2 = 9 \)

Combine the terms

Combine the calculated terms to express the polynomial in standard form:\[ 4x^2 - 12x + 9 \]

Key concepts

Special Product Formulas

To simplify polynomial multiplication, we often use special product formulas. These are patterns that provide shortcuts when expanding expressions like binomials. One of the most common is the square of a binomial formula, which states that \((a \pm b)^{2} = a^2 \pm 2ab + b^2\).
These formulas save time and help us avoid mistakes. Recognizing these patterns is key to mastering polynomial multiplication.
\( (2x-3)^{2} \) is one such expression. It fits the special product pattern, allowing us to easily expand and combine like terms.

Square of a Binomial

The square of a binomial formula is particularly important. We'll use it to simplify the given exercise. The formula is:
\((a \pm b)^{2} = a^2 \pm 2ab + b^2\)
When multiplying \( (2x-3)^{2} \, we follow these steps:

• Assign \((2x)\) to \(a\) and \(3\) to \(b\)
• Apply the formula: \((2x-3)^{2} = (2x)^{2} - 2(2x)(3) + 3^{2}\)
• Calculate each term: \( (2x)^{2} = 4x^{2}\), \-2(2x)(3) = -12x\), \(3^{2} = 9\)

Combining the terms, we get \ (4x^{2} - 12x + 9) \.

Standard Form

Standard form is a way to present polynomials in a clear and organized manner. It orders terms from highest to lowest degree.
For example, after expanding \ (2x-3)^{2} \ and simplifying, our polynomial becomes \ (4x^{2} - 12x + 9) \.
Notice the order:

• The \( x^{2} \) term comes first.
• Then the \( x \) term.
• Finally the constant term.

Presenting polynomials in standard form makes them easier to read and understand. This practice is crucial in algebra.