Fully factor the following polynomials. Explain your thought process with your solution.a. 𝑓(𝑥) = 2𝑥3 − 25𝑥2 + 53𝑥 − 30 [3 marks]b. 𝑓(𝑥) = 𝑥3 + 3𝑥 − 4 [3 marks]

a. To factor the polynomial 𝑓(𝑥) = 2𝑥³ − 25𝑥² + 53𝑥 − 30, we first look for any common factors. In this case, there are none.

Next, we look for patterns or use a method such as synthetic division to find the roots of the polynomial. Synthetic division is a shortcut method to divide a polynomial by a linear factor of the form x - a. It is easier and faster than traditional long division.

For this polynomial, we can use the Rational Root Theorem to find possible rational roots. The Rational Root Theorem states that any possible rational root, p/q, of a polynomial must be such that p is a factor of the constant term (in this case, -30) and q is a factor of the leading coefficient (in this case, 2).

By trying out these possible roots, we find that x = 2, x = 3, and x = 5 are roots of the polynomial.

Therefore, the fully factored form of the polynomial is 𝑓(𝑥) = 2(x - 2)(x - 3)(x - 5).

b. For the polynomial 𝑓(𝑥) = 𝑥³ + 3𝑥 − 4, we again look for common factors. There are none in this case.

We then look for possible roots using the Rational Root Theorem. Possible rational roots are factors of the constant term, -4.

By trying out these possible roots, we find that x = 1 and x = -2 are roots of the polynomial.

However, these roots only give us the factors (x - 1) and (x + 2). To find the remaining factor, we can divide the original polynomial by the product of these two factors.

Doing so gives us the remaining factor of x - 2.

Therefore, the fully factored form of the polynomial is 𝑓(𝑥) = (x - 1)(x + 2)(x - 2).