Question

Multiply \((x-2 y)(2 x-y)\). A. \(2 x^2+2 y^2\) B. \(2 x^2-5 x y+2 y^2\) C. \(2 x^2-2 y^2\) D. \(2 x^2+5 x y-2 y^2\)

Short answer

B. \(2 x^2-5 x y+2 y^2\)

Step-by-step solution

- Apply the Distributive Property

To multiply \(x-2y)((2x-y)\), apply the distributive property (FOIL method). This means distribute each term in the first binomial to every term in the second binomial.

- Multiply the Terms

First, multiply \(x\) by each term in the second binomial: \[ x \cdot 2x = 2x^2 \] \[ x \cdot (-y) = -xy \] Next, multiply \(-2y\) by each term in the second binomial: \[ -2y \cdot 2x = -4xy \] \[ -2y \cdot (-y) = 2y^2 \]

- Combine Like Terms

Now combine all the terms together: \[ 2x^2 - xy - 4xy + 2y^2 \] Combine the like terms \(-xy\) and \(-4xy\): \[ 2x^2 - 5xy + 2y^2 \]

- Identify the Correct Answer

The simplified expression is \(2x^2 - 5xy + 2y^2\), which matches option B.

Key concepts

distributive property

The distributive property is a fundamental algebraic principle that allows you to multiply a single term by two or more terms inside a set of parentheses. It is commonly represented as a(b + c) = ab + ac. This property ensures that you distribute the multiplication over the addition, breaking down complex expressions into simpler, manageable pieces. In our example, multiplying (x-2y)(2x-y), we distribute each term in the first binomial to every term in the second binomial. This turns a binomial multiplication into a series of simpler multiplication problems:

• x is multiplied by each term in the second binomial: \( x \cdot 2x = 2x^2 \) and \( x \cdot (-y) = -xy \)
• -2y is multiplied by each term in the second binomial: \( -2y \cdot 2x = -4xy \) and \( -2y \cdot (-y) = 2y^2 \)

FOIL method

When multiplying two binomials, the FOIL method is a handy mnemonic that stands for 'First', 'Outer', 'Inner', and 'Last' terms. It helps you remember the different parts of the binomials that need to be multiplied. Here's how it applies to our binomials (x-2y)(2x-y):

• First: Multiply the first terms in each binomial: \( x \cdot 2x = 2x^2 \)
• Outer: Multiply the outermost terms in each binomial: \( x \cdot (-y) = -xy \)
• Inner: Multiply the innermost terms in each binomial: \( -2y \cdot 2x = -4xy \)
• Last: Multiply the last terms in each binomial: \( -2y \cdot (-y) = 2y^2 \)

Remembering the FOIL method makes sure that no term is left out in the multiplication process, ensuring that all parts of the binomials are correctly multiplied.

like terms

After distributing and multiplying the terms, we end up with several terms that may look similar. These are called 'like terms' and can be combined to simplify the expression. Like terms are terms that have the same variable raised to the same power. In our solution (2x^2 - xy - 4xy + 2y^2), we combine the like terms involving x and y:

• The terms \( -xy \) and \( -4xy \) have the same variables (x and y), and can be combined to form \( -5xy \)

'simplifying the polynomial' is an important step to reach the final, simplest form of the expression so that it's easier to understand and use in further calculations. By combining like terms, we achieve a neatly simplified solution: (2x^2 - 5xy + 2y^2).

FOIL method

When multiplying two binomials, the FOIL method is a handy mnemonic that stands for 'First', 'Outer', 'Inner', and 'Last' terms. It helps you remember the different parts of the binomials that need to be multiplied. Here's how it applies to our binomials (x-2y)(2x-y):

• First: Multiply the first terms in each binomial: \( x \cdot 2x = 2x^2 \)
• Outer: Multiply the outermost terms in each binomial: \( x \cdot (-y) = -xy \)
• Inner: Multiply the innermost terms in each binomial: \( -2y \cdot 2x = -4xy \)
• Last: Multiply the last terms in each binomial: \( -2y \cdot (-y) = 2y^2 \)

Remembering the FOIL method makes sure that no term is left out in the multiplication process, ensuring that all parts of the binomials are correctly multiplied.

like terms

After distributing and multiplying the terms, we end up with several terms that may look similar. These are called 'like terms' and can be combined to simplify the expression. Like terms are terms that have the same variable raised to the same power. In our solution (2x^2 - xy - 4xy + 2y^2), we combine the like terms involving x and y:

• The terms \( -xy \) and \( -4xy \) have the same variables (x and y), and can be combined to form \( -5xy \)

'simplifying the polynomial' is an important step to reach the final, simplest form of the expression so that it's easier to understand and use in further calculations. By combining like terms, we achieve a neatly simplified solution: (2x^2 - 5xy + 2y^2).