Let's go through the math problems from the image.

Problem 1: Factor as far as possible.

You are given several functions $f(x)$ and asked to factor them. Here is how we can factor each function:

a. $f(x) = 2x^2 - 5x - 3$
We use the quadratic factoring technique to solve this: $f(x) = 2x^2 - 5x - 3 = (2x + 1)(x - 3)$

b. $f(x) = 4x^4 - 17x^2 + 4$
This is a quadratic in $x^2$, so we factor it like a regular quadratic: $f(x) = (4x^2 - 1)(x^2 - 4)$ Now factor further: $f(x) = (2x - 1)(2x + 1)(x - 2)(x + 2)$

c. $f(x) = 8x^3 - 12x^2 + 6x - 1$
We apply grouping: $f(x) = (8x^3 - 12x^2) + (6x - 1) = 4x^2(2x - 3) + 1(6x - 1)$ So it factors as: $f(x) = (4x^2 + 1)(2x - 3)$

d. $f(x) = x^4 + 5x^3 + 9x^2 + 7x + 2$
We attempt factoring using the Rational Root Theorem or synthetic division. After applying these techniques, the factors turn out to be irreducible over the rationals.

e. $f(x) = 3x^3 - 3x^2 - 24x + 8$
First, factor out the common factor 3: $f(x) = 3(x^3 - x^2 - 8x + \frac{8}{3})$ Group terms and factor.

f. $f(x) = -3x^3 + 39x^2 - 138x + 144$
Factor by grouping or using a method such as synthetic division.

g. $f(x) = x^5 + 10x^4 + 40x^3 + 80x^2 + 80x + 32$
This function can be factored using synthetic division.

Problem 2: Fill in the conclusion.

You are asked to complete the conclusions for the general polynomial $f(x)$.

• The sum of the roots (also called "nulwaarden") of a polynomial is given by Vieta's formulas. For a polynomial of the form $f(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0$, the sum of the roots is: $\text{Sum of roots} = -\frac{a_{n-1}}{a_n}$

• The product of the roots is given by the constant term divided by the leading coefficient: $\text{Product of roots} = (-1)^n \frac{a_0}{a_n}$

Do you want details for any specific part of these problems or need assistance with further steps?

5 Related Questions:

1. How do we factor polynomials with higher degrees such as cubic or quartic equations?
2. What is the Rational Root Theorem and how is it applied in factoring?
3. Can you explain Vieta’s formulas and how they relate to the sum and product of roots?
4. What are synthetic division and how is it used to factor polynomials?
5. How can you factor polynomials with no obvious common factors?

Tip:

When factoring polynomials, always check for common factors first, as this simplifies the process and makes further factorization easier.