Q(X)=2X^4+9X^3-5X^2-57X-45 find all the zeroes in exact form

To find the exact zeros of the polynomial \( Q(X) = 2X^4 + 9X^3 - 5X^2 - 57X - 45 \), we need to solve the equation \( Q(X) = 0 \).

### Step 1: Use Rational Root Theorem

The Rational Root Theorem suggests possible rational roots based on the factors of the constant term (-45) and the leading coefficient (2).

- Factors of -45: \( \pm 1, \pm 3, \pm 5, \pm 9, \pm 15, \pm 45 \)
- Factors of 2: \( \pm 1, \pm 2 \)

The possible rational roots are:

\[
\pm \frac{1}{1}, \pm \frac{1}{2}, \pm \frac{3}{1}, \pm \frac{3}{2}, \pm \frac{5}{1}, \pm \frac{5}{2}, \pm \frac{9}{1}, \pm \frac{9}{2}, \pm \frac{15}{1}, \pm \frac{15}{2}, \pm \frac{45}{1}, \pm \frac{45}{2}
\]

This gives the possible rational roots: \( \pm 1, \pm \frac{1}{2}, \pm 3, \pm \frac{3}{2}, \pm 5, \pm \frac{5}{2}, \pm 9, \pm \frac{9}{2}, \pm 15, \pm \frac{15}{2}, \pm 45, \pm \frac{45}{2} \).

### Step 2: Test Possible Rational Roots

Using synthetic division or direct substitution, we test possible roots. Let's start by testing \( X = -3 \):

\[
Q(-3) = 2(-3)^4 + 9(-3)^3 - 5(-3)^2 - 57(-3) - 45
\]
\[
Q(-3) = 2(81) + 9(-27) - 5(9) - 57(-3) - 45
\]
\[
Q(-3) = 162 - 243 - 45 + 171 - 45 = 0
\]

Thus, \( X = -3 \) is a root.

### Step 3: Polynomial Division

Now, divide \( Q(X) \) by \( (X + 3) \) to reduce the degree of the polynomial.

Perform synthetic division of \( 2X^4 + 9X^3 - 5X^2 - 57X - 45 \) by \( X + 3 \):

\[
\begin{array}{r|rrrrr}
-3 & 2 & 9 & -5 & -57 & -45 \\
   &   & -6 & -9 & 42 & 45 \\
\hline
   & 2 & 3 & -14 & -15 & 0 \\
\end{array}
\]

The quotient is \( 2X^3 + 3X^2 - 14X - 15 \).

### Step 4: Solve the Reduced Polynomial

Now, we solve the reduced polynomial \( 2X^3 + 3X^2 - 14X - 15 = 0 \).

Again, we can use the Rational Root Theorem. Possible rational roots for this cubic are:

\[
\pm 1, \pm \frac{1}{2}, \pm 3, \pm \frac{3}{2}, \pm 5, \pm \frac{5}{2}, \pm 15
\]

By testing \( X = -5 \):

\[
Q(-5) = 2(-5)^3 + 3(-5)^2 - 14(-5) - 15
\]
\[
= 2(-125) + 3(25) - 14(-5) - 15
\]
\[
= -250 + 75 + 70 - 15 = 0
\]

So, \( X = -5 \) is another root.

### Step 5: Divide Again

Now divide \( 2X^3 + 3X^2 - 14X - 15 \) by \( X + 5 \):

\[
\begin{array}{r|rrrr}
-5 & 2 & 3 & -14 & -15 \\
   &   & -10 & 35 & -105 \\
\hline
   & 2 & -7 & 0 & 0 \\
\end{array}
\]

The quotient is \( 2X^2 - 7 \).

### Step 6: Solve the Quadratic

The quadratic equation is \( 2X^2 - 7 = 0 \). Solving for \( X \):

\[
2X^2 = 7
\]
\[
X^2 = \frac{7}{2}
\]
\[
X = \pm \sqrt{\frac{7}{2}} = \pm \frac{\sqrt{14}}{2}
\]

### Step 7: List All Roots

The exact roots of the polynomial are:

\[
X = -3, \quad X = -5, \quad X = \pm \frac{\sqrt{14}}{2}
\]

Would you like more details or further explanation?

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Here are some related questions:

1. How do you use synthetic division in solving polynomials?
2. What is the Rational Root Theorem, and how is it applied?
3. How can you check if a solution is correct in polynomial equations?
4. What are complex roots, and when do they occur?
5. How can factoring help in solving higher-degree polynomials?

**Tip**: Always check your polynomial division results by multiplying back to ensure accuracy!

Discover the exact zeroes of the polynomial Q(X) = 2X^4 + 9X^3 - 5X^2 - 57X - 45 using the Rational Root Theorem and synthetic division. This comprehensive guide is ideal for students in Grades 10-12, covering key algebraic techniques and solving higher-degree polynomial equations.

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