Question

Multiply the polynomials using the FOIL method. Express your answer as a single polynomial in standard form. $$ (-3 x+4)(x-2) $$

Short answer

The product of \(( -3x + 4)( x - 2)\) is \( -3x^2 + 10x - 8\).

Step-by-step solution

- Identify the Terms

Identify the terms in each binomial. The first binomial is \( -3x + 4 \) and the second binomial is \( x - 2 \).

- Multiply the First Terms

Multiply the first terms of each binomial: \( (-3x) \) and \( x \). The product is \( -3x \cdot x = -3x^2 \).

- Multiply the Outer Terms

Multiply the outer terms of each binomial: \( (-3x) \) and \( -2 \). The product is \( -3x \cdot (-2) = 6x \).

- Multiply the Inner Terms

Multiply the inner terms of each binomial: \( 4 \) and \( x \). The product is \( 4 \cdot x = 4x \).

- Multiply the Last Terms

Multiply the last terms of each binomial: \( 4 \) and \( -2 \). The product is \( 4 \cdot (-2) = -8 \).

- Add the Products

Combine all the products: \( -3x^2 \), \( 6x \), \( 4x \), and \( -8 \).

- Simplify the Expression

Combine like terms to express the answer as a single polynomial in standard form: \( -3x^2 + 6x + 4x - 8 = -3x^2 + 10x - 8 \).

Key concepts

Polynomial Multiplication

Polynomial multiplication involves distributing each term in the first polynomial to every term in the second polynomial. This concept helps us find the product of two polynomials and is a foundational skill in algebra. Imagine having multiple baskets and you need to distribute items equally to each basket. Each item from one basket interacts with every item from another. Similarly, each term in the first polynomial interacts with each term in the second polynomial. In simpler terms, it’s like using the distributive property multiple times. When we solve polynomials using the FOIL (First, Outer, Inner, Last) method, we follow specific steps that break down these interactions systematically.

Binomial Products

Binomial products are the result of multiplying two binomials. A binomial is an algebraic expression with two terms, like \(-3x + 4\) and \(x - 2\). When you multiply these two, you use the FOIL method:

• First: Multiply the first terms in each binomial. For our example, \(-3x \cdot x = -3x^2\).
• Outer: Multiply the outer terms. Here, it means \(-3x \cdot (-2) = 6x\).
• Inner: Multiply the inner terms. This gives us \(4 \cdot x = 4x\).
• Last: Multiply the last terms in each binomial. So, \(4 \cdot (-2) = -8\).

By organizing our calculation this way, the process becomes more structured and understandable. Each step tackles a specific pair of terms, ensuring we don’t miss any combination. Combining these intermediate products results in the final polynomial.

Simplifying Expressions

Once we've obtained the products from applying the FOIL method to our polynomials, the next essential step is simplifying the expression. Simplifying here means combining like terms to convert our multiple products into a single polynomial.

• Combining like terms: Terms with the same exponent on the variable are considered like terms. In our example, we have \(-3x^2\) from the first terms, \(6x\) from the outer terms, \(4x\) from the inner terms, and \(-8\) from the last terms.
• Combine like terms to simplify: The intermediate products are then added together to form one single expression in standard form. For our problem, the combining yields \(-3x^2 + 6x + 4x - 8\). Simplifying this, we get \-3x^2 + 10x - 8\.

Standard form is often written as \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. This form helps in understanding and further manipulating the polynomial. By ensuring that we combine like terms methodically, we reach an accurate and streamlined expression that is easier to work with in subsequent mathematical operations.