Question

Multiply the polynomials using the special product formulas. Express your answer as a single polynomial in standard form. $$ (x-5)^{2} $$

Short answer

The result is \[ x^2 - 10x + 25 \].

Step-by-step solution

Identify the special product formula

Recognize that the expression \( (x-5)^{2} \) is a perfect square trinomial. The formula for the square of a binomial \( (a-b)^{2} \) is \[ (a-b)^{2} = a^2 - 2ab + b^2 \].

Apply the formula with identified values

Substitute \( a = x \) and \( b = 5 \) into the formula. \( (x-5)^{2} \) becomes \[ x^2 - 2 \times x \times 5 + 5^2 \].

Simplify each term

Calculate each term in the expanded expression. \( x^2 \) remains \( x^2 \, \-2 \times x \times 5 \) simplifies to \-10x \ , and \( 5^2 \) is \ 25. \ So, the expression becomes: \ x^2 - 10x + 25.

Write the final expression in standard form

Combine the terms to write the polynomial in standard form as \[ x^2 - 10x + 25 \].

Key concepts

Polynomial Multiplication

Understanding polynomial multiplication is crucial for working with algebraic expressions. When multiplying polynomials, you need to multiply each term in one polynomial by each term in the other.
To multiply polynomials:

• Distribute each term from the first polynomial to every term in the second polynomial.
• Combine like terms.

In our example, we deal with the special case of a squared binomial \((x-5)^2\). Here, we’re not multiplying two different polynomials but rather taking a single binomial and multiplying it by itself. This brings us to an important special product formula.

Special Product Formulas

Special product formulas help simplify multiplication of certain binomials without expanding everything manually each time. One key formula is the square of a binomial.
When we square a binomial (in the form \(a-b\) or \(a+b\)), we use the formula:

• \( (a-b)^2 = a^2 - 2ab + b^2 \)

This formula helps quickly break down the multiplication process.
In our example, \( (x-5)^2 \), we identify \( a=x \) and \( b=5 \). Applying the formula leads to:
\[ (x-5)^2 = x^2 - 2(x)(5) + 5^2 \]
Which further simplifies to:
\

Standard Form Polynomial

Polynomials are often written in standard form, which means arranging the terms in descending order of their exponents. This makes it easier to see the structure and compare polynomials.
The standard form of a polynomial ensures that:

• The terms are ordered from highest to lowest degree.
• Each term includes coefficients and variables.

For our example, \[ x^2 - 10x + 25 \], we start with the term having the largest exponent \(x^2\) and follow it with \(-10x\) then finally the constant \(25\).
Writing polynomials in standard form is a common step after performing operations like multiplication.
By consistently ordering terms, you make it clearer and more manageable for further mathematical operations.