Factorise following polynomials into linear factors. Show all working to receive full marks.

P(x) = 5x^3 - 4x^2 - 52x -48

To factorize the polynomial \( P(x) = 5x^3 - 4x^2 - 52x - 48 \) into linear factors, we need to perform the following steps:

### Step 1: Find a potential root using the Rational Root Theorem
The Rational Root Theorem suggests that any rational root of the polynomial \( P(x) \) is a factor of the constant term (-48) divided by a factor of the leading coefficient (5). So the possible rational roots are:

\[
\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 16, \pm 24, \pm 48, \pm \frac{1}{5}, \pm \frac{2}{5}, \pm \frac{3}{5}, \pm \frac{4}{5}, \pm \frac{6}{5}, \pm \frac{8}{5}, \pm \frac{12}{5}, \pm \frac{16}{5}, \pm \frac{24}{5}, \pm \frac{48}{5}
\]

We will test these potential roots by substituting them into \( P(x) \).

### Step 2: Test potential roots
Let's start by testing \( x = -2 \).

\[
P(-2) = 5(-2)^3 - 4(-2)^2 - 52(-2) - 48
\]
\[
P(-2) = 5(-8) - 4(4) + 104 - 48
\]
\[
P(-2) = -40 - 16 + 104 - 48 = 0
\]

Since \( P(-2) = 0 \), \( x = -2 \) is a root. This means \( x + 2 \) is a factor of \( P(x) \).

### Step 3: Perform synthetic division
We now divide \( P(x) \) by \( x + 2 \) using synthetic division.

The coefficients of \( P(x) \) are: \( 5, -4, -52, -48 \).

\[
\begin{array}{r|rrrr}
-2 & 5 & -4 & -52 & -48 \\
   &   & -10 & 28 & 48 \\
\hline
   & 5 & -14 & -24 & 0 \\
\end{array}
\]

The quotient is \( 5x^2 - 14x - 24 \) and the remainder is 0, confirming \( x + 2 \) is a factor.

So we have:
\[
P(x) = (x + 2)(5x^2 - 14x - 24)
\]

### Step 4: Factor the quadratic \( 5x^2 - 14x - 24 \)
We now need to factorize \( 5x^2 - 14x - 24 \). To do this, we look for two numbers whose product is \( 5 \times -24 = -120 \) and whose sum is \( -14 \).

The numbers are \( 6 \) and \( -20 \). Therefore, we can rewrite the quadratic:

\[
5x^2 - 20x + 6x - 24
\]

Factor by grouping:

\[
= 5x(x - 4) + 6(x - 4)
\]

Factor out \( (x - 4) \):

\[
= (5x + 6)(x - 4)
\]

### Step 5: Write the fully factorized form
Thus, the fully factorized form of \( P(x) \) is:

\[
P(x) = (x + 2)(5x + 6)(x - 4)
\]

### Final Answer:
\[
\text{The polynomial } P(x) = 5x^3 - 4x^2 - 52x - 48 \text{ can be factorized as } (x + 2)(5x + 6)(x - 4).
\]

Would you like any further details or have any questions?

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Here are 8 related questions you might find helpful:
1. How can you factor polynomials with four terms by grouping?
2. What is the difference between synthetic division and long division of polynomials?
3. How does the Rational Root Theorem help in factorizing polynomials?
4. What are the steps to factor a quadratic polynomial?
5. How can you verify that your factorization of a polynomial is correct?
6. Can you always factor cubic polynomials into linear factors?
7. How do you determine if a cubic polynomial has a complex root?
8. What is the Factor Theorem, and how does it apply to polynomial factorization?

**Tip:** When factorizing, always test your factors by multiplying them back together to ensure you return to the original polynomial.

Learn how to factorize the polynomial 5x^3 - 4x^2 - 52x - 48 into linear factors with detailed steps and examples. This problem involves polynomial factorization using the Rational Root Theorem and synthetic division. Suitable for advanced high school students.

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