Questions: Factor the trinomial completely. 28 x^3 + 51 x^2 + 11 x Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. 28 x^3 + 51 x^2 + 11 x = (Factor completely.) B. The polynomial is prime.

Factor the trinomial completely.
28 x^3 + 51 x^2 + 11 x

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. 28 x^3 + 51 x^2 + 11 x = (Factor completely.)
B. The polynomial is prime.

Transcript text: 02 Intermediate Algebra (2253-1) k: IA 13.3 on Objective A Objective B Objective C tive B: Factor Out a GCF Before Factoring a Trinomial of the Form $a x^{2}+b x+c$ UR TURN \#1 0 of 1 Point Factor the trinomial completely. \[ 28 x^{3}+51 x^{2}+11 x \] Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. $28 x^{3}+51 x^{2}+11 x=\square$ (Factor completely.) $\square$ B. The polynomial is prime. Help me solve this View an example Ask my instructor

Solution

Solution Steps

Step 1: Identify the Greatest Common Factor (GCF)

First, identify the GCF of the terms $ 28x^{3} $, $ 51x^{2} $, and $ 11x $. The GCF of the coefficients $ 28 $, $ 51 $, and $ 11 $ is $ 1 $. The GCF of the variable terms $ x^{3} $, $ x^{2} $, and $ x $ is $ x $. Therefore, the GCF of the entire expression is $ x $.

Step 2: Factor Out the GCF

Factor out the GCF $ x $ from the trinomial: \[ 28x^{3} + 51x^{2} + 11x = x(28x^{2} + 51x + 11) \]

Step 3: Factor the Remaining Quadratic Expression

Now, factor the quadratic expression $ 28x^{2} + 51x + 11 $. To do this, find two numbers that multiply to $ 28 \times 11 = 308 $ and add to $ 51 $. The numbers $ 44 $ and $ 7 $ satisfy this condition because $ 44 \times 7 = 308 $ and $ 44 + 7 = 51 $.

Rewrite the middle term using these numbers: \[ 28x^{2} + 44x + 7x + 11 \]

Group the terms: \[ (28x^{2} + 44x) + (7x + 11) \]

Factor out the GCF from each group: \[ 4x(7x + 11) + 1(7x + 11) \]

Factor out the common binomial factor $ (7x + 11) $: \[ (7x + 11)(4x + 1) \]

Step 4: Write the Final Factored Form

Combine the GCF $ x $ with the factored quadratic expression: \[ 28x^{3} + 51x^{2} + 11x = x(7x + 11)(4x + 1) \]