Question

Multiply the polynomials using the FOIL method. Express your answer as a single polynomial in standard form. $$ (2 x+7)(x+5) $$

Short answer

The product is 2x^2 + 17x + 35

Step-by-step solution

- Multiply the First terms

Multiply the first terms in each binomial: a. First terms: a1. 2x a2. x a3. 2x * x = 2x^2

- Multiply the Outer terms

Multiply the outer terms in each binomial: a. Outer terms: a1. 2x a2. 5 a3. 2x * 5 = 10x

- Multiply the Inner terms

Multiply the inner terms in each binomial: a. Inner terms: a1. 7 a2. x a3. 7 * x = 7x

- Multiply the Last terms

Multiply the last terms in each binomial: a. Last terms: a1. 7 a2. 5 a3. 7 * 5 = 35

- Combine all the products

Combine all the terms you have found so far: a. 2x^2 + 10x + 7x + 35

- Simplify to standard form

Simplify the expression by combining like terms: a. Combine 10x and 7x b. 2x^2 + 17x + 35 This is the product of the given binomials in standard form.

Key concepts

Multiplying Polynomials

Multiplying polynomials involves finding the product of each term in one polynomial with each term in the other polynomial. In this exercise, you were given two binomials to multiply using the FOIL method.

The FOIL method is a handy technique specific to multiplying two binomials. FOIL stands for:

• First
• Outer
• Inner
• Last

These terms represent the order in which you multiply the parts of the binomials.

Let's break this down further using the example \( (2x + 7)(x + 5) \). Here's how you use the FOIL method step-by-step:

• First: Multiply the first terms of each binomial: \( 2x \times x = 2x^2 \)
• Outer: Multiply the outer terms: \( 2x \times 5 = 10x \)
• Inner: Multiply the inner terms: \( 7 \times x = 7x \)
• Last: Multiply the last terms of each binomial: \( 7 \times 5 = 35 \)

Combine all these products: \( 2x^2 + 10x + 7x + 35 \).
Finally, add the like terms \( 10x \) and \( 7x \) to get the product of the binomials in standard form: \( 2x^2 + 17x + 35 \).

Standard Form

In algebra, polynomials are often written in what is called 'standard form.' Standard form means that the terms are arranged in descending order of their exponents. For example, the polynomial \( 2x^2 + 17x + 35 \) is in standard form because the term with the highest exponent (\(x^2\)) comes first, followed by the term with the next highest exponent (\(x\)), and then the constant term.

More specifically, the general form of writing a polynomial in standard form is:

• highest exponent term
• next highest exponent term
• constant if there is one

For instance, in the final result \( 2x^2 + 17x + 35 \):

• \(2x^2\) is the term with the highest exponent (2)
• \(17x\) is the next highest term with an exponent of 1
• \(35\) is the constant term

This standard form helps to communicate your answer clearly and makes it easier for anyone reading it to understand the polynomial structure right away. So, always arrange your final polynomial result in descending order of their exponents.

Binomials

A binomial is a polynomial with exactly two terms. For instance, in the exercise given \( (2x + 7)(x + 5) \), each pair has two terms separated by a plus or minus sign.

To identify binomials more easily, look for:

• Two distinct terms
• Each term can have coefficients and/or variables
• They can be added or subtracted

Using the given example, the binomials are \( 2x + 7 \) and \( x + 5 \). Here, both binomials follow the rule of having exactly two terms.

The binomial structure allows us to use methods like FOIL to multiply them efficiently. Each operation covers every pairing of terms from each binomial, ensuring no terms are missed.

Understanding binomials is crucial for mastering polynomial operations, as they often serve as building blocks for more complex polynomial equations. Identifying and working with binomials will deepen your understanding of algebraic expressions and their manipulations.