https://file.aiquickdraw.com/mathbot/file/17300059445845xjk4q15.jpg

Let's solve each part of the question step-by-step.

Given:
\[
P(x) = x^2 + 2x + 3 \quad \text{and} \quad Q(x) = 4x^2 + 5x + 6
\]

### Part (a): \( 3P(x) \)
To find \( 3P(x) \), multiply each term in \( P(x) \) by 3:
\[
3P(x) = 3(x^2 + 2x + 3) = 3x^2 + 6x + 9
\]

### Part (b): \( P(x) + Q(x) \)
To find \( P(x) + Q(x) \), add the corresponding terms in \( P(x) \) and \( Q(x) \):
\[
P(x) + Q(x) = (x^2 + 2x + 3) + (4x^2 + 5x + 6)
\]
Combine like terms:
\[
P(x) + Q(x) = (x^2 + 4x^2) + (2x + 5x) + (3 + 6) = 5x^2 + 7x + 9
\]

### Part (c): \( P(x) - 2Q(x) \)
To find \( P(x) - 2Q(x) \), first find \( 2Q(x) \) by multiplying \( Q(x) \) by 2:
\[
2Q(x) = 2(4x^2 + 5x + 6) = 8x^2 + 10x + 12
\]
Now subtract \( 2Q(x) \) from \( P(x) \):
\[
P(x) - 2Q(x) = (x^2 + 2x + 3) - (8x^2 + 10x + 12)
\]
Combine like terms:
\[
P(x) - 2Q(x) = (x^2 - 8x^2) + (2x - 10x) + (3 - 12) = -7x^2 - 8x - 9
\]

### Part (d): \( P(x)Q(x) \)
To find \( P(x)Q(x) \), multiply \( P(x) \) and \( Q(x) \):
\[
P(x)Q(x) = (x^2 + 2x + 3)(4x^2 + 5x + 6)
\]
Use distribution (or the FOIL method for each pair of terms):
\[
P(x)Q(x) = x^2(4x^2 + 5x + 6) + 2x(4x^2 + 5x + 6) + 3(4x^2 + 5x + 6)
\]
Expanding each term:
1. \( x^2 \cdot 4x^2 = 4x^4 \)
2. \( x^2 \cdot 5x = 5x^3 \)
3. \( x^2 \cdot 6 = 6x^2 \)
4. \( 2x \cdot 4x^2 = 8x^3 \)
5. \( 2x \cdot 5x = 10x^2 \)
6. \( 2x \cdot 6 = 12x \)
7. \( 3 \cdot 4x^2 = 12x^2 \)
8. \( 3 \cdot 5x = 15x \)
9. \( 3 \cdot 6 = 18 \)

Now combine like terms:
\[
P(x)Q(x) = 4x^4 + (5x^3 + 8x^3) + (6x^2 + 10x^2 + 12x^2) + (12x + 15x) + 18
\]
\[
= 4x^4 + 13x^3 + 28x^2 + 27x + 18
\]

---

Let me know if you would like further explanation on any part.

### Follow-up Questions:
1. What is the general approach to multiplying two polynomials?
2. How do you determine the degree of a polynomial resulting from multiplication?
3. Can you explain why combining like terms is essential in polynomial operations?
4. How would the result change if we had \( P(x) \) and \( Q(x) \) with higher degrees?
5. How do you interpret the terms and coefficients in each polynomial product?

### Tip:
When multiplying polynomials, it’s helpful to write each step carefully to avoid missing any terms, especially with higher degrees.

If P(x) = x^2 + 2x + 3 and Q(x) = 4x^2 + 5x + 6, find in simplest form:

a. 3P(x)
b. P(x) + Q(x)
c. P(x) - 2Q(x)
d. P(x)Q(x)

Learn to simplify polynomial expressions using given functions P(x) = x^2 + 2x + 3 and Q(x) = 4x^2 + 5x + 6. This example covers addition, subtraction, multiplication, and scalar multiplication of polynomials. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation