Question

Multiply the polynomials using the FOIL method. Express your answer as a single polynomial in standard form. $$ (x+3)(x+5) $$

Short answer

\( x^2 + 8x + 15 \)

Step-by-step solution

Identify the given polynomials

The given polynomials are \( (x+3) \) and \( (x+5) \).

Apply the FOIL Method

FOIL stands for First, Outer, Inner, Last. Multiply these terms accordingly: \[ (x+3)(x+5) = (x \cdot x) + (x \cdot 5) + (3 \cdot x) + (3 \cdot 5) \]

Multiply the 'First' terms

First multiply the first terms in each binomial. Here, \( x \cdot x = x^2 \)

Multiply the 'Outer' terms

Next, multiply the outer terms in the binomials. Here, \( x \cdot 5 = 5x \)

Multiply the 'Inner' terms

Now, multiply the inner terms in the binomials. Here, \( 3 \cdot x = 3x \)

Multiply the 'Last' terms

Finally, multiply the last terms in each binomial. Here, \( 3 \cdot 5 = 15 \)

Combine all terms

Combine all the multiplied terms: \[ x^2 + 5x + 3x + 15 \]

Combine like terms

Combine the like terms, \( 5x \) and \( 3x \), to simplify the expression. \[ x^2 + 8x + 15 \]

Key concepts

Polynomial Multiplication

Polynomial multiplication involves multiplying each term in one polynomial by each term in another polynomial and then combining the results. In our example, we are given two binomials: \( (x+3) \) and \( (x+5) \). To multiply these, we will use the FOIL method, a convenient technique particularly for binomials. Each term in one binomial must be multiplied by each term in the second binomial. This results in four products that we then add together.

Binomials

A binomial is a polynomial with exactly two terms. In the example, \( (x+3) \) and \( (x+5) \) are binomials because each has two terms. When multiplying binomials, the FOIL method helps to simplify the process by remembering to multiply:

• F: First terms
• O: Outer terms
• I: Inner terms
• L: Last terms

This ensures that no term is left out during the multiplication process.

Combining Like Terms

After using the FOIL method to multiply the binomials, we usually get multiple terms that need to be simplified. Like terms are terms that have the same variable raised to the same power. In our solution, the result after using FOIL is \[ x^2 + 5x + 3x + 15 \]. We see that \[ 5x \] and \[ 3x \] are like terms because they both have \`x\` as the variable. Combining like terms involves adding their coefficients to simplify the expression. Hence, \[ 5x + 3x = 8x \], giving the simplified result: \[ x^2 + 8x + 15 \].

Standard Form

After multiplying and simplifying the terms, the final expression should be written in standard form. The standard form of a polynomial is to write it with terms in descending order of their exponents. For our example, the final expression \[ x^2 + 8x + 15 \] is already in standard form with the terms in order of \[ x^2 \] to the constant. Writing polynomials in standard form makes them easier to read and understand.