p(x)=x4−4x3−24x2+20x+7
are 1
 and 7
. The other two zeros are real, but irrational.
The smaller is 
equation editorEquation Editor , and the larger is 
equation editorEquation Editor .
Hint: Divide p
 by two linear factors or by one quadratic factor.

Given the polynomial:

\[
p(x) = x^4 - 4x^3 - 24x^2 + 20x + 7
\]

we know that two of the zeros are \( x = 1 \) and \( x = 7 \). To find the remaining two zeros, we can start by performing polynomial division.

### Step 1: Divide \( p(x) \) by \( (x - 1)(x - 7) \)

This suggests we divide \( p(x) \) by the product of these linear factors:

\[
(x - 1)(x - 7) = x^2 - 8x + 7
\]

Thus, our task is to divide \( p(x) \) by \( x^2 - 8x + 7 \) to find the quotient. If we let this quotient be \( q(x) \), we have:

\[
p(x) = (x^2 - 8x + 7) \cdot q(x)
\]

### Step 2: Polynomial Division of \( p(x) \) by \( x^2 - 8x + 7 \)

1. **First Division Term**: Divide \( x^4 \) by \( x^2 \) to get \( x^2 \).
2. Multiply \( x^2 \) by \( x^2 - 8x + 7 \), giving:
   \[
   x^4 - 8x^3 + 7x^2
   \]
3. Subtract from \( p(x) \):
   \[
   (x^4 - 4x^3 - 24x^2 + 20x + 7) - (x^4 - 8x^3 + 7x^2) = 4x^3 - 31x^2 + 20x + 7
   \]

4. **Next Division Term**: Divide \( 4x^3 \) by \( x^2 \) to get \( 4x \).
5. Multiply \( 4x \) by \( x^2 - 8x + 7 \), giving:
   \[
   4x^3 - 32x^2 + 28x
   \]
6. Subtract:
   \[
   (4x^3 - 31x^2 + 20x + 7) - (4x^3 - 32x^2 + 28x) = x^2 - 8x + 7
   \]

7. **Final Division Term**: Divide \( x^2 \) by \( x^2 \) to get \( 1 \).
8. Multiply \( 1 \) by \( x^2 - 8x + 7 \), yielding \( x^2 - 8x + 7 \).
9. Subtract:
   \[
   (x^2 - 8x + 7) - (x^2 - 8x + 7) = 0
   \]

Thus, the quotient is:

\[
q(x) = x^2 + 4x + 1
\]

### Step 3: Solve for the Remaining Zeros Using \( q(x) = x^2 + 4x + 1 \)

Now, we set \( q(x) = 0 \) and solve:

\[
x^2 + 4x + 1 = 0
\]

Using the quadratic formula:

\[
x = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{-4 \pm \sqrt{16 - 4}}{2} = \frac{-4 \pm \sqrt{12}}{2} = \frac{-4 \pm 2\sqrt{3}}{2} = -2 \pm \sqrt{3}
\]

### Conclusion

The remaining two zeros are:

\[
x = -2 + \sqrt{3} \quad \text{and} \quad x = -2 - \sqrt{3}
\]

The **smaller zero** is \( -2 - \sqrt{3} \), and the **larger zero** is \( -2 + \sqrt{3} \).

Would you like more details on this solution or have any questions?

Here are some related questions to deepen your understanding:

1. How can we confirm that \( x = 1 \) and \( x = 7 \) are indeed zeros of \( p(x) \)?
2. What is the geometric interpretation of real and irrational roots?
3. How do we determine the end behavior of a polynomial like \( p(x) \)?
4. How does the degree of the polynomial impact the nature and number of its roots?
5. How can synthetic division help in dividing polynomials with known roots?

**Tip:** Remember, the quadratic formula is a powerful tool for finding zeros, especially when they are irrational roots!

The polynomial p(x) = x^4 − 4x^3 − 24x^2 + 20x + 7 has two known zeros at 1 and 7. The other two zeros are real but irrational. The smaller is Equation Editor, and the larger is Equation Editor. Hint: Divide p by two linear factors or by one quadratic factor.

Explore how to find the real and irrational roots of the polynomial x^4 − 4x^3 − 24x^2 + 20x + 7. This detailed solution walks through polynomial division, solving a quadratic equation, and identifying irrational roots. Suitable for high school students (Grades 9-12).

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