Question

Multiply the polynomials using the special product formulas. Express your answer as a single polynomial in standard form. $$ (3 x-4)^{2} $$

Short answer

The polynomial is \( 9x^2 - 24x + 16 \) in standard form.

Step-by-step solution

Identify the formula to use

Recognize that the expression \( (3x - 4)^2 \) fits the square of a binomial formula which is \( (a - b)^2 = a^2 - 2ab + b^2 \). Here, \( a = 3x \) and \( b = 4 \).

Apply the formula

Substitute \( a = 3x \) and \( b = 4 \) into the formula \( (a - b)^2 = a^2 - 2ab + b^2 \): \[ (3x - 4)^2 = (3x)^2 - 2(3x)(4) + (4)^2 \]

Calculate each term

Compute each term separately: \[ (3x)^2 = 9x^2 \] \[ -2(3x)(4) = -24x \] \[ (4)^2 = 16 \]

Combine the terms

Combine all the calculated terms to express the polynomial in standard form: \[ (3x - 4)^2 = 9x^2 - 24x + 16 \]

Key concepts

square of a binomial

A binomial is an algebraic expression containing two terms separated by a plus (+) or minus (-) sign. The square of a binomial refers to the expression when a binomial is multiplied by itself. For example, the square of (3x - 4)is written as (3x - 4)^2.
To multiply a binomial by itself, you can use the special product formula: (a - b)^2 = a^2 - 2ab + b^2.
Here, a and b can be any terms.
In the exercise, 3x is our a and 4 is our b. Applying the formula, we get: (3x - 4)^2 = (3x)^2 - 2(3x)(4) + (4)^2.
This breakdown simplifies the process, making it easier to handle multiplying binomials.

polynomial multiplication

Polynomial multiplication involves multiplying each term of one polynomial with every term of another polynomial. In our specific case of the binomial (3x - 4) squared, we use a special product formula that simplifies the multiplication process.
The general approach to polynomial multiplication includes these steps:

• Multiply each term in the first polynomial by each term in the second polynomial.
• Combine like terms (terms with the same variables raised to the same power).

For our binomial, we followed this approach by using the special product formula: (3x - 4)^2 = (3x)^2 - 2(3x)(4) + (4)^2.
Simplifying each term separately, we obtained the individual parts: (3x)^2 = 9x^2,-2(3x)(4) = -24x, and (4)^2 = 16.
This step-by-step multiplication ensures accuracy and clarity.

standard form of polynomials

The standard form of a polynomial is writing it so that terms are ordered from the highest degree to the lowest degree. Each term should be combined if they are like terms (having the same variables raised to the same power).
In our example, after multiplying (3x - 4)^2, we get three terms: 9x^2, -24x, and 16.
We then combine these terms to write the polynomial in its standard form: 9x^2 - 24x + 16.
This organized format is important for simplifying expressions and solving equations in algebra.
So, whenever you find the product of polynomials, always finish by arranging the resulting polynomial in standard form to make it easy to read and work with.