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Q-2 Multiply the following polynomials. 


1 𝑝(𝑥) = 3𝑥^2− 4𝑥 + 7, 𝑞(𝑥) = 7𝑥^2− 2𝑥 − 1

 2 𝑝(𝑥) = 𝑥^2+ 3𝑥 + 5, 𝑞(𝑥) = 2𝑥^2+ 𝑥 − 9

To multiply the two given polynomials, we will use the distributive property, often referred to as the "FOIL" method for binomials. Here’s the process for both pairs of polynomials:

### 1. Multiplying \( p(x) = 3x^2 - 4x + 7 \) and \( q(x) = 7x^2 - 2x - 1 \):

We distribute each term in \( p(x) \) to each term in \( q(x) \), and then combine like terms.

\[
p(x) \cdot q(x) = (3x^2 - 4x + 7)(7x^2 - 2x - 1)
\]

Now multiply each term:

- \( 3x^2 \cdot 7x^2 = 21x^4 \)
- \( 3x^2 \cdot (-2x) = -6x^3 \)
- \( 3x^2 \cdot (-1) = -3x^2 \)

- \( -4x \cdot 7x^2 = -28x^3 \)
- \( -4x \cdot (-2x) = 8x^2 \)
- \( -4x \cdot (-1) = 4x \)

- \( 7 \cdot 7x^2 = 49x^2 \)
- \( 7 \cdot (-2x) = -14x \)
- \( 7 \cdot (-1) = -7 \)

Now combine all the terms:

\[
21x^4 - 6x^3 - 3x^2 - 28x^3 + 8x^2 + 4x + 49x^2 - 14x - 7
\]

Combine like terms:

\[
21x^4 - 34x^3 + 54x^2 - 10x - 7
\]

Thus, the product of the two polynomials is:

\[
p(x) \cdot q(x) = 21x^4 - 34x^3 + 54x^2 - 10x - 7
\]

---

### 2. Multiplying \( p(x) = x^2 + 3x + 5 \) and \( q(x) = 2x^2 + x - 9 \):

Similar to the previous example, distribute each term:

\[
p(x) \cdot q(x) = (x^2 + 3x + 5)(2x^2 + x - 9)
\]

Now multiply each term:

- \( x^2 \cdot 2x^2 = 2x^4 \)
- \( x^2 \cdot x = x^3 \)
- \( x^2 \cdot (-9) = -9x^2 \)

- \( 3x \cdot 2x^2 = 6x^3 \)
- \( 3x \cdot x = 3x^2 \)
- \( 3x \cdot (-9) = -27x \)

- \( 5 \cdot 2x^2 = 10x^2 \)
- \( 5 \cdot x = 5x \)
- \( 5 \cdot (-9) = -45 \)

Now combine all the terms:

\[
2x^4 + x^3 - 9x^2 + 6x^3 + 3x^2 - 27x + 10x^2 + 5x - 45
\]

Combine like terms:

\[
2x^4 + 7x^3 + 4x^2 - 22x - 45
\]

Thus, the product of the two polynomials is:

\[
p(x) \cdot q(x) = 2x^4 + 7x^3 + 4x^2 - 22x - 45
\]

---

Let me know if you have any questions or need further details!

### Relative Questions:
1. How do you simplify the product of two polynomials?
2. What is the significance of combining like terms in polynomial multiplication?
3. Can you explain the FOIL method for binomials?
4. How do you approach multiplying polynomials of higher degrees?
5. What are common mistakes to avoid when multiplying polynomials?

### Tip:
When multiplying polynomials, keep track of signs carefully to avoid errors when combining terms.

Q-2 Multiply the following polynomials: 1. p(x) = 3x^2 − 4x + 7, q(x) = 7x^2 − 2x − 1 2. p(x) = x^2 + 3x + 5, q(x) = 2x^2 + x − 9

Learn how to multiply polynomials using the distributive property, including step-by-step solutions for p(x) = 3x^2 − 4x + 7 with q(x) = 7x^2 − 2x − 1, and p(x) = x^2 + 3x + 5 with q(x) = 2x^2 + x − 9. This tutorial is suitable for high school students (Grades 9-12) and covers key algebraic concepts.

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