Question

represent the quadratic polynomial \\(2x^2 + x - 6\\) using algebra tiles and determine the equivalent factored form.
the number of zero pairs needed to model this polynomial is dropdown.
the equivalent factored form is dropdown.

Explanation:

Analyze the quadratic polynomial

We are given the quadratic polynomial \(2x^2 + x - 6\). To represent this polynomial using Algebra Tiles in a rectangular grid, we need to arrange two \(+x^2\) tiles, some \(x\) tiles (both positive and negative), and six \(-1\) tiles into a perfect rectangle.

Determine the required zero pairs

To factor \(2x^2 + x - 6\) using Factoring Trinomials, we look for two numbers that multiply to \(a \cdot c = 2 \cdot (-6) = -12\) and add up to \(b = 1\).
These two numbers are \(4\) and \(-3\) because:

$$ 4 \cdot (-3) = -12 $$

$$ 4 + (-3) = 1 $$

This means we need to split the middle term \(+x\) into \(4x - 3x\).
In terms of Algebra Tiles, we start with one \(+x\) tile. To get \(4\) positive \(x\) tiles and \(3\) negative \(x\) tiles, we must add zero pairs of \(x\) tiles (each pair consists of one \(+x\) and one \(-x\) tile).
Since we need \(4\) positive \(x\) tiles and we only have \(1\) positive \(x\) tile, we must add \(3\) more positive \(x\) tiles along with \(3\) negative \(-x\) tiles.
Thus, the number of zero pairs of \(x\) tiles needed is \(3\).

Find the equivalent factored form

Using the split middle term, we rewrite and factor the polynomial by grouping:

$$ 2x^2 + 4x - 3x - 6 $$

Group the terms:

$$ (2x^2 + 4x) - (3x + 6) $$

Factor out the greatest common factor from each group:

$$ 2x(x + 2) - 3(x + 2) $$

Factor out the common binomial \((x + 2)\):

$$ (2x - 3)(x + 2) $$

This represents the dimensions of the rectangle formed by the algebra tiles, which gives the equivalent factored form.

Answer:

Represent the quadratic polynomial \(2x^2 + x - 6\) using algebra tiles and determine the equivalent factored form.
The number of zero pairs needed to model this polynomial is <blank>3</blank>.
The equivalent factored form is <blank>\((2x - 3)(x + 2)\)</blank>.