Problem Analysis:
The given problem involves polynomial division. We are dividing the polynomial \( 3x^4 - 9x^3 + x^2 + 15x + p \) by \( 3x^2 - 5 \). The division process is partially shown, and we need to determine the value of \( p \) such that the remainder is zero.
Step-by-Step Solution:
#### 1.
Setup the Division:
The dividend is \( 3x^4 - 9x^3 + x^2 + 15x + p \), and the divisor is \( 3x^2 - 5 \).
#### 2.
First Step of Division:
- Divide the leading term of the dividend (\( 3x^4 \)) by the leading term of the divisor (\( 3x^2 \)):
\[
\frac{3x^4}{3x^2} = x^2
\]
- Multiply the entire divisor \( 3x^2 - 5 \) by \( x^2 \):
\[
x^2 \cdot (3x^2 - 5) = 3x^4 - 5x^2
\]
- Subtract this result from the original dividend:
\[
(3x^4 - 9x^3 + x^2 + 15x + p) - (3x^4 - 5x^2) = -9x^3 + 6x^2 + 15x + p
\]
#### 3.
Second Step of Division:
- Divide the leading term of the new polynomial (\( -9x^3 \)) by the leading term of the divisor (\( 3x^2 \)):
\[
\frac{-9x^3}{3x^2} = -3x
\]
- Multiply the entire divisor \( 3x^2 - 5 \) by \( -3x \):
\[
-3x \cdot (3x^2 - 5) = -9x^3 + 15x
\]
- Subtract this result from the current polynomial:
\[
(-9x^3 + 6x^2 + 15x + p) - (-9x^3 + 15x) = 6x^2 + p
\]
#### 4.
Third Step of Division:
- Divide the leading term of the new polynomial (\( 6x^2 \)) by the leading term of the divisor (\( 3x^2 \)):
\[
\frac{6x^2}{3x^2} = 2
\]
- Multiply the entire divisor \( 3x^2 - 5 \) by \( 2 \):
\[
2 \cdot (3x^2 - 5) = 6x^2 - 10
\]
- Subtract this result from the current polynomial:
\[
(6x^2 + p) - (6x^2 - 10) = p + 10
\]
#### 5.
Final Remainder:
The final remainder is \( p + 10 \). For the division to be exact (i.e., no remainder), the remainder must be zero:
\[
p + 10 = 0
\]
#### 6.
Solve for \( p \):
\[
p = -10
\]
Final Answer:
\[
\boxed{-10}
\]
Parent Tip: Review the logic above to help your child master the concept of algebraic long division worksheet.
