Question

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

Short answer

Option C: \((x-y)(2x+y)\)

Step-by-step solution

Identify the quadratic expression

The given quadratic expression is \(2 x^2 - x y - y^2\). The goal is to factorize it into a product of two binomials.

Set up the factorization form

Assume the factorized form is \((ax + by)(cx + dy)\). Then expand this product to match the original quadratic expression.

Expand the assumed form

Expand \((ax + by)(cx + dy)\):\[(ax + by)(cx + dy) = acx^2 + (ad + bc)xy + bdy^2\]

Equate coefficients

Compare the expanded form with \(2 x^2 - x y - y^2\) to find coefficients:\[ac = 2\]\[ad + bc = -1\]\[bd = -1\]

Solve for coefficients

Possible coefficient pairs:\(a = 2\), \(c = 1\); and \(b = -1\), \(d = 1\). Verify by substituting:\[(2x - y)(x + y) \]

Verify the solution

Expand \((2x - y)(x + y)\):\[(2x - y)(x + y) = 2x^2 + 2xy - xy - y^2 = 2x^2 - xy - y^2\]Results match the original expression, verifying the correct factorization.

Key concepts

polynomial factorization

Polynomial factorization is the process of breaking down a polynomial into products of simpler polynomials. This is often done to simplify expressions or solve equations. For quadratic equations, the goal is to write the equation as a product of two binomials.

For example, if we want to factor the quadratic expression given in the exercise: \(2x^2 - xy - y^2\). We start by assuming it can be expressed as \((ax + by)(cx + dy) \) and then expand and compare coefficients to find the values for a, b, c, and d that satisfy the equation.

algebraic expressions

An algebraic expression is a combination of numbers, variables, and operation symbols (such as +, -, *, and /). These can be as simple as a single variable or number, or as complex as multi-term polynomials.

In the given exercise, our algebraic expression is a quadratic one: \(2x^2 - xy - y^2\). To solve such expressions, we follow the steps of identifying, factoring, expanding, and verifying. Each step involves careful manipulation of the algebraic expression to ensure accuracy.

quadratic equations

Quadratic equations are a type of polynomial equation that have the form \(ax^2 + bx + c = 0\), where a, b, and c are constants. These equations are essential in algebra and appear in various math problems.

In the given exercise, the quadratic equation is \(2x^2 - xy - y^2 = 0\). To solve it by factoring, we convert it to \((2x - y)(x + y)\), thus breaking it down into two simpler equations.

• The first factor: \(2x - y = 0\)
• The second factor: \(x + y = 0\)

This approach makes solving the equation more manageable and helps in finding the values for x and y that satisfy the equation.