Question

factor the trinomial, or state that the trinomial is prime.
5x² - 14x - 3
select the correct choice below and, if necessary, fill in the answer.
a. 5x² - 14x - 3 = (factor completely.)
b. the polynomial is prime.

Explanation:

Step1: Multiply leading - coefficient and constant

For the trinomial \(ax^{2}+bx + c=5x^{2}-14x - 3\), \(a = 5\), \(b=-14\), \(c = - 3\). Calculate \(a\times c=5\times(-3)=-15\).

Step2: Find two numbers that multiply to \(ac=-15\) and add to \(b=-14\)

The two numbers are \(-15\) and \(1\) since \(-15\times1=-15\) and \(-15 + 1=-14\).

Step3: Rewrite the middle - term

Rewrite \(-14x\) as \(-15x+x\). So, \(5x^{2}-14x - 3=5x^{2}-15x+x - 3\).

Step4: Group the terms

\((5x^{2}-15x)+(x - 3)\).

Step5: Factor out the greatest common factor from each group

From \(5x^{2}-15x\), factor out \(5x\) to get \(5x(x - 3)\), and from \(x - 3\) it remains as is. So, \(5x(x - 3)+1(x - 3)\).

Step6: Factor out the common binomial factor

Factor out \((x - 3)\) to get \((x - 3)(5x+1)\).

Answer:

A. \(5x^{2}-14x - 3=(x - 3)(5x + 1)\)