Question

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

Short answer

The expanded polynomial is \( x^2 - 8x + 16 \).

Step-by-step solution

Identify the Special Product Formula

Recognize that the given expression \((x-4)^2\) can be expanded using the square of a binomial formula: \((a-b)^2 = a^2 - 2ab + b^2 \). Here, \(a = x\) and \(b = 4\).

Apply the Formula

Substitute \(a\) and \(b\) into the formula: \( (x-4)^2 = x^2 - 2(x)(4) + 4^2 \).

Simplify Each Term

Calculate each part of the expression: \( x^2 \) stays the same, \( -2(x)(4) \) simplifies to \(-8x\), and \(4^2\) simplifies to \16\.

Combine the Terms

Combine all terms to write the polynomial in standard form: \[ x^2 - 8x + 16 \].

Key concepts

Binomial Expansion

Binomial expansion is a powerful tool in algebra that lets you expand expressions that are raised to a power. A binomial is an algebraic expression that contains two terms, like \(a - b\) or \(x + y\).
When you have these binomials raised to a power, you can use specific formulas to expand them efficiently. One of these formulas is the square of a binomial:
\((a - b)^2 = a^2 - 2ab + b^2\).
This formula saves you time because it gives you the result directly without needing to multiply everything out manually.
For example, in the exercise, we have \(x - 4\) and need to square it. By identifying \(a\) and \(b\), where \(a = x\) and \(b = 4\), and then substituting them into the formula, we get \((x - 4)^2 = x^2 - 2(x)(4) + 4^2\).
Now we just need to simplify each part to find the expanded form.

Polynomial Multiplication

Polynomial multiplication involves multiplying two or more polynomials together. When you multiply polynomials, you use each term in the first polynomial to multiply each term in the second polynomial.
This might sound complicated, but special product formulas can make it much easier.
In our exercise, we used \(a - b\)'s squared formula. After we applied the binomial expansion formula, we performed polynomial multiplication as follows: \(x^2 - 2(x)(4) + 4^2\).
Breaking it down, this becomes:

• First: \(x \times x = x^2\)
• Second: -2 times the product of \(x \times 4 = -8x\)
• Third: the square of \(4 = 16\)

Now, by simplifying each term and combining them, we move to the next essential concept, which is writing our result in standard form.

Standard Form

Once we have expanded and simplified our polynomial, the final step is expressing it in standard form. The standard form of a polynomial arranges the terms in decreasing order of their powers.
For example, let's look at our result: \(x^2 - 8x + 16\).
In standard form, the highest power term \(x^2\) comes first, followed by the linear term \(-8x\), and finally, the constant \(16\).
Standard form makes it easier to understand and work with polynomials, especially when solving equations or graphing.
This organization helps you quickly identify the degree of the polynomial and the leading coefficient, which are essential for many algebraic processes.
So, after finishing the expansion and simplification process, always arrange your polynomial in standard form to ensure clarity and ease of use in future calculations.