To determine the correct grouping of the trinomial [tex]\(6x^2 - x - 5\)[/tex], we need to express it in a way that separates the middle term, [tex]\(-x\)[/tex], into two terms whose coefficients sum to the coefficient of the middle term and whose product is the product of the coefficients of the first and last terms of the trinomial.
First, recall the trinomial given:
[tex]\[6x^2 - x - 5\][/tex]
Now, consider each of the given choices:
1. [tex]\(6x^2 - 9x + 8x - 5\)[/tex]
2. [tex]\(6x^2 - x + 30x - 5\)[/tex]
3. [tex]\(6x^2 - 4x + 33x - 5\)[/tex]
4. [tex]\(6x^2 - 6x + 5x - 5\)[/tex]
5. [tex]\(6x^2 - 5x + 4x - 5\)[/tex]
Let's analyze each option based on the requirement mentioned:
Option 1: [tex]\(6x^2 - 9x + 8x - 5\)[/tex]
- Here, [tex]\(-9x + 8x\)[/tex] results in [tex]\(-x\)[/tex].
- Then, we check the multiplication: [tex]\( (6x^2) \cdot (-5) = 6x^2 \cdot (-5) = -30x^2 \)[/tex].
- And for the terms we use, [tex]\(-9x \cdot 8x = -72x^2\)[/tex].
This option matches because:
-9x + 8x = -x,
and they balance out, giving us back the original coefficients.
Option 2: [tex]\(6x^2 - x + 30x - 5\)[/tex]
- Here, [tex]\(-x + 30x\)[/tex] results in [tex]\( 29x\)[/tex], which does not match the middle term of [tex]\( -x\)[/tex]. Thus this choice is incorrect.
Option 3: [tex]\(6x^2 - 4x + 33x - 5\)[/tex]
- Here, [tex]\(-4x + 33x\)[/tex] results in [tex]\( 29x\)[/tex], which again does not match the middle term of [tex]\( -x\)[/tex]. Thus this is also incorrect.
Option 4: [tex]\(6x^2 - 6x + 5x - 5\)[/tex]
- Here, [tex]\(-6x + 5x\)[/tex] results in [tex]\( -1x\)[/tex], and using same coefficients add up correctly.
However, let's check the multiplication:
[tex]\((6x^2 \cdot -5= -30x^2 )\)[/tex]
breaking as above tells us incorrect factors led.
-thus we must reject this choice as
Option 5: [tex]\(6x^2 - 5x + 4x - 5\)[/tex]
Here, [tex]\(-5x + 4x\)[/tex] results in [tex]\(-1x \)[/tex],, but the factors splitting - not adding to correct multiplication as original trinomial, hence incorrect
Upon reviewing all options only :
The correct grouping for the given trinomial [tex]\(6x^2 - x - 5\)[/tex] is:
[tex]\[6x^2 - 9x + 8x - 5\][/tex]
Therefore, the correct answer is:
[tex]\[6x^2 - 9x + 8x - 5\][/tex]
