Question

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

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

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

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

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

Explanation:

Step1: Recall Distributive Property

The distributive property (also known as the distributive law of multiplication over addition/subtraction) states that for any expressions \(a\), \(b\), and \(c\), \(a(b + c)=ab+ac\) and \(a(b - c)=ab - ac\). Here, we have \((2x^{2}+4x - 7)(x - 3)\), which can be thought of as \(a(b - c)\) where \(a = 2x^{2}+4x - 7\), \(b=x\), and \(c = 3\). So, by the distributive property, \(a(b - c)=ab - ac\), which translates to \((2x^{2}+4x - 7)(x)+(2x^{2}+4x - 7)(- 3)\) (since \(ab - ac=ab+a(-c)\)).

Step2: Analyze Other Options

• Option A: It has \(+3(2x^{2}+4x - 7)\), but according to the distributive property, it should be \(-3(2x^{2}+4x - 7)\) (because it's \(x-3=x+(-3)\)), so A is incorrect.
• Option C: It only multiplies \(-3x\) with the trinomial, but we need to distribute both \(x\) and \(-3\) over the trinomial, so C is incorrect.
• Option D: It has \(x(2x^{2}+4x - 7)-3\), but we need to distribute \(-3\) over the trinomial \(2x^{2}+4x - 7\), not just subtract 3. So D is incorrect.

Answer:

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