if $f(-5)=0$, what are all the factors of the function $f(x)=x^{3}-19x + 30$? use the remainder…

if $f(-5)=0$, what are all the factors of the function $f(x)=x^{3}-19x + 30$? use the remainder theorem.\n$(x - 2)(x + 5)(x - 3)$\n$(x + 2)(x - 5)(x + 3)$\n$(x - 2)(x + 5)$\n$(x + 2)(x - 5)$

if $f(-5)=0$, what are all the factors of the function $f(x)=x^{3}-19x + 30$? use the remainder theorem.\n$(x - 2)(x + 5)(x - 3)$\n$(x + 2)(x - 5)(x + 3)$\n$(x - 2)(x + 5)$\n$(x + 2)(x - 5)$

Answer

Explanation:

Step1: Apply Remainder Theorem

Since (f(-5)=0), then ((x + 5)) is a factor of (f(x)=x^{3}-19x + 30).

Step2: Perform polynomial long - division

Divide (x^{3}-19x + 30) by (x + 5). We have (x^{3}-19x + 30=(x + 5)(x^{2}-5x+6)).

Step3: Factor the quadratic

Factor (x^{2}-5x + 6) as ((x - 2)(x - 3)) using the formula (x^{2}+bx + c=(x - x_1)(x - x_2)) where (x_1,x_2) are roots of (x^{2}+bx + c = 0) and for (x^{2}-5x + 6=0), (x=\frac{5\pm\sqrt{25 - 24}}{2}=\frac{5\pm1}{2}), so (x_1 = 2,x_2=3).

Step4: Write all factors

So (f(x)=(x - 2)(x + 5)(x - 3)).

Answer:

A. ((x - 2)(x + 5)(x - 3))