Question

Which of the following is equivalent to \(2 x^2-6 x-8\) ? A) \(2(x-4)(x+1)\) B) \(3(x+4)(x-1)\) C) \(2(x-3)(x+2)\) D) \(3(x-4)(x-2)\)

Short answer

The correct answer is Option A, $2(x-4)(x+1)$, which is equivalent to $2x^2-6x-8$.

Step-by-step solution

Identify the given expression

The given expression is \(2x^2 - 6x - 8\).

Expand Option A

Expand Option A, \( 2(x-4)(x+1)\), using the FOIL method: First: \(x \times x = x^2\) Outer: \(x \times 1 = x\) Inner: \(-4 \times x = -4x\) Last: \(-4 \times 1 = -4\) Now, combine the terms: \(x^2 + x - 4x - 4\) which simplifies to \(x^2 - 3x - 4\). Finally, multiply by 2 (as there's a 2 in front of the parentheses): \(2(x^2 - 3x - 4) = 2x^2 - 6x - 8\). Since this matches the original expression, we can confirm that Option A is the correct answer. However, it's good to double-check the other options to ensure that they do not simplify to the given expression.

Expand Option B

Expand Option B, \( 3(x+4)(x-1)\), using the FOIL method: First: \(x \times x = x^2\) Outer: \(x \times -1 = -x\) Inner: \(4 \times x = 4x\) Last: \(4 \times -1 = -4\) Now, combine the terms: \(x^2 - x + 4x - 4\) which simplifies to \(x^2 + 3x - 4\). Finally, multiply by 3 (as there's a 3 in front of the parentheses): \(3(x^2 + 3x - 4) = 3x^2 + 9x - 12\). This does not match the original expression, so Option B can be eliminated.

Expand Option C

Expand Option C, \( 2(x-3)(x+2)\), using the FOIL method: First: \(x \times x = x^2\) Outer: \(x \times 2 = 2x\) Inner: \(-3 \times x = -3x\) Last: \(-3 \times 2 = -6\) Now, combine the terms: \(x^2 + 2x - 3x - 6\) which simplifies to \(x^2 - x - 6\). Finally, multiply by 2 (as there's a 2 in front of the parentheses): \(2(x^2 - x - 6) = 2x^2 - 2x - 12\). This does not match the original expression, so Option C can be eliminated.

Expand Option D

Expand Option D, \( 3(x-4)(x-2)\), using the FOIL method: First: \(x \times x = x^2\) Outer: \(x \times -2 = -2x\) Inner: \(-4 \times x = -4x\) Last: \(-4 \times -2 = 8\) Now, combine the terms: \(x^2 - 2x - 4x + 8\) which simplifies to \(x^2 - 6x + 8\). Finally, multiply by 3 (as there's a 3 in front of the parentheses): \(3(x^2 - 6x + 8) = 3x^2 - 18x + 24\). This does not match the original expression, so Option D can be eliminated.

Identify the correct answer

Since we have expanded and verified that only Option A, \(2(x-4)(x+1)\), results in the given expression \(2x^2 - 6x - 8\), we can conclude that \(2(x-4)(x+1)\) is equivalent to \(2x^2 - 6x - 8\). Therefore, the correct answer is Option A.

Key concepts

Polynomial Factoring

Understanding the concept of polynomial factoring is essential for solving various mathematical problems, especially when dealing with quadratic expressions. Factoring is the process of breaking down a complex expression into simpler factors that, when multiplied together, give back the original expression. It is a reverse process of expanding and is particularly handy when you are seeking the roots of a polynomial equation or simplifying algebraic fractions.

For instance, factoring the quadratic polynomial \(2x^2 - 6x - 8\) involves finding two binomial expressions that, when multiplied, give the original quadratic polynomial. A correct approach often begins with looking for two numbers that multiply to the product of the leading coefficient (2) and the constant term (-8), which in this case is -16, while also adding up to the middle coefficient (-6). Upon finding such a pair of numbers (-8 and +2), the quadratic can be factored as \(2(x - 4)(x + 1)\), revealing the simpler components of the polynomial.

FOIL Method

The FOIL method is a technique used to multiply two binomials. The acronym FOIL stands for First, Outer, Inner, Last, which represent the order in which you multiply the terms of the binomials. Let's break down the process:

• First: Multiply the first terms in each binomial.
• Outer: Multiply the outer terms in the product.
• Inner: Multiply the inner terms.
• Last: Multiply the last terms of the binomials.

After performing these operations, you combine like terms to simplify the expression. For example, to multiply \(2(x - 4)(x + 1)\), you would first apply the FOIL method to \(x - 4\) and \(x + 1\):

First: \(x \times x = x^2\)
Outer: \(x \times 1 = x\)
Inner: \(—4 \times x = —4x\)
Last: \(—4 \times 1 = —4\)

Then, \(x^2 - 4x + x - 4\) simplifies to \(x^2 - 3x - 4\). Multiplying by the leading coefficient 2, as indicated, gives the original quadratic expression \(2x^2 - 6x - 8\). The FOIL method not only helps in expansion but also in verifying factored forms of polynomial expressions.

Quadratic Expressions

Quadratic expressions are second-degree polynomials, typically written in the standard form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). They can represent various phenomena, such as projectile motion in physics or profit functions in economics. The graph of a quadratic expression is a parabola, which opens upwards if \(a > 0\) and downwards if \(a < 0\).

Factoring quadratic expressions is a valuable skill that provides insight into their properties, such as the x-intercepts of their graph, the vertex of the parabola, and the axis of symmetry. For the SAT Math Problem Solving exercise example of \(2x^2 - 6x - 8\), factoring it into \(2(x - 4)(x + 1)\) helps to identify that the expression equals zero when \(x = 4\) or \(x = -1\), which are the x-intercepts of the corresponding parabola. Furthermore, understanding and manipulating these expressions play a critical role in solving quadratic equations, which is a vital component of high school math curriculums.