Question

use a geometric model to factor ( 3x^2 + 4x + 1 ) by following these steps: step 1: model the trinomial by placing tiles to form a rectangle in the top left corner of the product section. step 2: drag tiles to the sections labeled factor 1 and factor 2 to represent the factors. step 3: what is the factored form of this trinomial? ( circ (x - 1)(3x + 1) ) ( circ (3x + 1)(x + 1) ) ( circ (3x - 1)(x + 1) ) ( circ (3x - 1)(x - 1) ) correct! cnx

Explanation:

Step1: Recall factoring trinomials

To factor \(3x^2 + 4x + 1\), we can also use the method of expanding the options. Let's expand each option:

• For \((x - 1)(3x + 1)\):

Expand using FOIL: \(x(3x)+x(1)-1(3x)-1(1)=3x^2 + x - 3x - 1 = 3x^2 - 2x - 1\), which is not the given trinomial.

• For \((3x + 1)(x + 1)\):

Expand using FOIL: \(3x(x)+3x(1)+1(x)+1(1)=3x^2 + 3x + x + 1 = 3x^2 + 4x + 1\), which matches the given trinomial.

• For \((3x - 1)(x + 1)\):

Expand using FOIL: \(3x(x)+3x(1)-1(x)-1(1)=3x^2 + 3x - x - 1 = 3x^2 + 2x - 1\), not matching.

• For \((3x - 1)(x - 1)\):

Expand using FOIL: \(3x(x)-3x(1)-1(x)+1(1)=3x^2 - 3x - x + 1 = 3x^2 - 4x + 1\), not matching.

Step2: Confirm the correct factor

From the expansion above, only \((3x + 1)(x + 1)\) gives the trinomial \(3x^2 + 4x + 1\).

Answer:

\((3x + 1)(x + 1)\) (corresponding to the option: \((3x + 1)(x + 1)\))