Question

Multiply the polynomials using the FOIL method. Express your answer as a single polynomial in standard form. $$ (x-5)(x-1) $$

Short answer

The product is \( x^2 - 6x + 5 \).

Step-by-step solution

- Apply the FOIL Method

Use the FOIL method to multiply the binomials. FOIL stands for First, Outer, Inner, Last.Multiply the first terms: \( x \times x = x^2 \)Multiply the outer terms: \( x \times -1 = -x \)Multiply the inner terms: \( -5 \times x = -5x \)Multiply the last terms: \( -5 \times -1 = 5 \)

- Combine Like Terms

Add up all the products from the FOIL method: \( x^2 + (-x) + (-5x) + 5 \). Combine the like terms: \( -x - 5x = -6x \)

- Write in Standard Form

Combine everything to form the final polynomial: \( x^2 - 6x + 5 \)

Key concepts

Polynomial Multiplication

When multiplying polynomials, you are essentially expanding the expression to include every possible product between the terms of the polynomials.
One of the most common techniques is the FOIL method, which stands for First, Outer, Inner, Last.
The FOIL method is particularly useful for multiplying two binomials (polynomials with two terms). In our example, we are multiplying \( (x-5)(x-1) \).
Here’s what each part means:

• First: Multiply the first terms of each binomial together, which is \( x \times x = x^2 \).
• Outer: Multiply the outer terms together, which means \( x \times -1 = -x \).
• Inner: Multiply the inner terms together, in this case, \( -5 \times x = -5x \).
• Last: Multiply the last terms together, which is \( -5 \times -1 = 5 \).

Doing this step by step for each pair of terms ensures that you don’t miss any products, which is essential for accurate polynomial multiplication.

Combining Like Terms

After using the FOIL method, you typically end up with several terms, some of which are 'like terms.'
Like terms are terms that have the same variable raised to the same power.
In our example: \( x^2 + (-x) + (-5x) + 5 \), the like terms are \( -x \) and \( -5x \).
To combine them, simply add their coefficients together.

• \(-x \) and \(-5x \) are like terms. When combined, you get \(-6x \).

This makes the expression \( x^2 - 6x + 5 \).
Combining like terms simplifies the polynomial, making it easier to understand and work with.

Standard Form of a Polynomial

The standard form of a polynomial is a way of writing the polynomial where terms are arranged in descending order by their degree (the exponent of the variable).
Degrees go from highest to lowest: \ ax^n + bx^{n-1} + ... + c \.

• For example, \( x^2 - 6x + 5 \) is in standard form, where \( x^2 \) is the highest degree term, followed by \( x \), and then the constant term (5).
• Writing polynomials in this way helps in clearly understanding the leading term, which often determines the polynomial's essential characteristics, such as its degree.

The standard form is very important for polynomial comparison and further algebraic operations.
Make sure to always check that your final expression is ordered correctly.