Question

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

Short answer

The product is \(2x^2 + 17x + 35\).

Step-by-step solution

Identify the Terms

Separate each polynomial into its terms. The first polynomial \(-x - 5\) has terms \(-x\) and \(-5\). The second polynomial \(-2x - 7\) has terms \(-2x\) and \(-7\).

Apply FOIL Method

FOIL stands for First, Outer, Inner, Last. Multiply the terms as follows:1. First: \(-x \times -2x = 2x^2\)2. Outer: \(-x \times -7 = 7x\)3. Inner: \(-5 \times -2x = 10x\)4. Last: \(-5 \times -7 = 35\)

Combine Like Terms

Add the results from the FOIL steps together:\[2x^2 + 7x + 10x + 35\]Combine the like terms \(7x\) and \(10x\) to get:\[2x^2 + 17x + 35\]

Write in Standard Form

The final polynomial in standard form is:\[2x^2 + 17x + 35\]

Key concepts

Polynomial Multiplication

Polynomial multiplication is a method used to find the product of two polynomials. When multiplying polynomials, each term in the first polynomial must be multiplied by each term in the second polynomial. In this exercise, we use the FOIL (First, Outer, Inner, Last) method to multiply the polynomials \((-x - 5)\) and \((-2x - 7)\). The FOIL method allows us to systematically multiply each term to ensure that no terms are omitted. Let's break that down:

• The first terms: \(-x \times -2x = 2x^{2}\)
• The outer terms: \(-x \times -7 = 7x\)
• The inner terms: \(-5 \times -2x = 10x\)
• The last terms: \(-5 \times -7 = 35\)

After multiplying, you'll combine these products to form the new polynomial before simplifying.

Standard Form

Standard form for polynomials means expressing the polynomial with terms in descending order by degree. After using the FOIL method and finding the products, the next step is to write the polynomial in standard form. For the polynomial \[2x^{2} + 7x + 10x + 35\], start by combining like terms. Here, these are terms with the same variable raised to the same power:

• \[7x + 10x = 17x\]

Now combine everything together to have it in standard form:

\[2x^{2} + 17x + 35\].

This arrangement makes it easier to understand the polynomial and is the widely accepted format for presenting your final answer.

Combining Like Terms

Combining like terms is an essential step in simplifying polynomials. Like terms are terms that have the same variable raised to the same power. In the step-by-step process, after using the FOIL method, you get multiple terms that need to be combined.

For instance, after FOIL-ing \((-x - 5)\) and \((-2x - 7)\), you end up with:

\[2x^{2} + 7x + 10x + 35\]

The terms \[7x\] and \[10x\] are like terms because they both contain the variable \[x\] raised to the first power. Adding these together gives you \[17x\].

Finally, you include all the combined terms to get the polynomial in its most simplified version, which in this case, becomes:\[2x^{2} + 17x + 35\]