Question

question 7 of 10
which expression is equivalent to ((x^2 - 2x)(3x^2 + 4x - 7))?

a. (x^2(3x^2 + 4x) - 2x(4x - 7))

b. (x^2(3x^2 + 4x - 7) + 2x(3x^2 + 4x - 7))

c. (x^2(3x^2 + 4x - 7) - 2x(3x^2 + 4x - 7))

d. (x^2(3x^2 + 4x - 7) - 2x)

Explanation:

Step1: Recall the distributive property (FOIL for polynomials)

The distributive property states that \(a(b + c)=ab+ac\), and for subtraction, \(a(b - c)=ab - ac\). Here, we have \((x^{2}-2x)(3x^{2}+4x - 7)\), which can be seen as \(a(b + c - d)\) where \(a=x^{2}-2x\), \(b = 3x^{2}\), \(c = 4x\), \(d=7\), or more simply, using the distributive property for two - term multiplication: \((m - n)p=mp - np\), where \(m=x^{2}\), \(n = 2x\), and \(p=(3x^{2}+4x - 7)\).

Step2: Apply the distributive property

Using \((m - n)p=mp - np\) with \(m=x^{2}\), \(n = 2x\), and \(p=(3x^{2}+4x - 7)\), we get:
\((x^{2}-2x)(3x^{2}+4x - 7)=x^{2}(3x^{2}+4x - 7)-2x(3x^{2}+4x - 7)\)

Let's check the other options:

• Option A: The second term is \(-2x(4x - 7)\) instead of \(-2x(3x^{2}+4x - 7)\), so it's incorrect.
• Option B: It has a plus sign between the two terms (\(x^{2}(3x^{2}+4x - 7)+2x(3x^{2}+4x - 7)\)) instead of a minus sign, so it's incorrect.
• Option D: The second term is just \(-2x\) instead of \(-2x(3x^{2}+4x - 7)\), so it's incorrect.

Answer:

C. \(x^{2}(3x^{2}+4x - 7)-2x(3x^{2}+4x - 7)\)