Question

Multiply the polynomials using the FOIL method. Express your answer as a single polynomial in standard form. $$ (-2 x-3)(3-x) $$

Short answer

2x^2 - 3x - 9

Step-by-step solution

Identify the terms in each binomial

The first binomial is ewline (-2x - 3) and the second binomial is (3 - x)

Apply FOIL method - First terms

Multiply the first terms: ewline (-2x) * 3 = -6x

Apply FOIL method - Outer terms

Multiply the outer terms: ewline (-2x) * (-x) = 2x^2

Apply FOIL method - Inner terms

Multiply the inner terms: ewline (-3) * 3 = -9

Apply FOIL method - Last terms

Multiply the last terms: ewline (-3) * (-x) = 3x

Combine all the products

Combine all the terms obtained from FOIL: ewline -6x + 2x^2 - 9 + 3x

Simplify the expression

Combine like terms to express the polynomial in standard form: ewline 2x^2 - 3x - 9

Key concepts

polynomial multiplication

Polynomial multiplication involves expanding expressions that have more than one term combined by addition or subtraction. In this example, we're multiplying two binomials using the FOIL method. It stands for First, Outer, Inner, and Last, representing a structured way to ensure every term from each polynomial is multiplied correctly.

Understanding the steps is crucial, because multiplying polynomials forms the foundation for many advanced algebra topics. When multiplying, remember to:

• Multiply each term systematically
• Apply the distributive property correctly
• Combine all obtained products

This structured method enhances clarity, ensuring no terms are missed and the resulting polynomial is accurate.

binomials

A binomial is a polynomial with exactly two terms. In our exercise, the binomials are \((-2x - 3)\) and \( (3 - x) \). Understanding the structure of binomials is key to using the FOIL method effectively.

Binomials can have variables, constants, or both, and they often look like sums or differences between two distinct parts. Here are some examples of binomials:

• \(2y + 5\)
• \(-4x - 7\)
• \(a - b\)

Recognizing binomials quickly helps in identifying the correct terms to multiply when using FOIL or any other polynomial multiplication technique.

combining like terms

After applying the FOIL method, the next step is combining like terms, to simplify the polynomial to its standard form. Like terms have the same variable raised to the same power.

For instance, in our example:

• \((-6x)\) and \((3x)\) are like terms because they both contain \(x\)
• \((-9)\) is a constant term, isolated
• \((2x^2)\) stands alone as it's the highest degree term

By combining like terms correctly, we get the polynomial in standard form, ensuring clarity and correctness. This simplifies solving equations and further manipulations significantly.

algebraic expressions

Algebraic expressions are combinations of numbers, variables, and operations. They form the basic building blocks of algebra. In the given exercise, expressions like \((-2x - 3)\) and \( (3 - x) \) are algebraic expressions.

Key points to understand about algebraic expressions:

• They can be simplified using algebraic rules
• They can be combined through addition, subtraction, multiplication, and division
• Variables in expressions represent unknown values that can vary

Grasping how to manipulate algebraic expressions is fundamental for solving not just polynomial problems, but also equations and inequalities in algebra. This allows for deeper understanding and flexibility in problem-solving.