Questions: QUESTION 22 If possible, factor the polynomial completely. If a polynomial cannot be factored, state that it is prime. 14 x^2-49 x-28

Transcript text: QUESTION 22 If possible, factor the polynomial completely. If a polynomial cannot be factored, state that it is prime. \[ 14 x^{2}-49 x-28 \]

Solution

Solution Steps

To factor the polynomial \(14x^2 - 49x - 28\), we can use the method of factoring by grouping or the quadratic formula to find the roots and then express the polynomial as a product of its factors. If the polynomial cannot be factored using integers, we will state that it is prime.

Step 1: Identify the Polynomial

We start with the polynomial \(14x^2 - 49x - 28\).

Step 2: Factor the Polynomial

To factor the polynomial, we can express it in the form \(a(b)(c)\). The polynomial can be factored as follows: \[ 14x^2 - 49x - 28 = 7(x - 4)(2x + 1) \]

Step 3: Verify the Factorization

We can verify the factorization by expanding \(7(x - 4)(2x + 1)\): \[ 7(x - 4)(2x + 1) = 7[(2x^2 + x - 8x - 4)] = 7(2x^2 - 7x - 4) = 14x^2 - 49x - 28 \] This confirms that the factorization is correct.