Polynomial Operations

In this lesson, we will explore the operations you can perform on polynomials, including addition, subtraction, multiplication, and division. Polynomials are expressions consisting of variables and coefficients, involving terms in the form of $a_nx^n + a_{n-1}x^{n-1} + \ldots + a_{1}x + a_{0}$.

Example: Adding and Multiplying Polynomials

Let's consider two polynomials:

$P(x) = 3x^{2} + 2x + 1$
$Q(x) = 5x^{2} + 4x + 3$

Step 1: Addition of Polynomials

To add $P(x)$ and $Q(x)$, we combine like terms:

$R(x) = P(x) + Q(x)$
$R(x) = (3x^{2} + 2x + 1) + (5x^{2} + 4x + 3)$
$R(x) = (3x^{2} + 5x^{2}) + (2x + 4x) + (1 + 3)$
$R(x) = 8x^{2} + 6x + 4$

Step 2: Multiplication of Polynomials

Now, let's multiply $P(x)$ and $Q(x)$:

$S(x) = P(x) \cdot Q(x)$
$S(x) = (3x^{2} + 2x + 1)(5x^{2} + 4x + 3)$

We will use the distributive property (FOIL method for binomials):

1. $3x^{2} \cdot 5x^{2} = 15x^{4}$
2. $3x^{2} \cdot 4x = 12x^{3}$
3. $3x^{2} \cdot 3 = 9x^{2}$
4. $2x \cdot 5x^{2} = 10x^{3}$
5. $2x \cdot 4x = 8x^{2}$
6. $2x \cdot 3 = 6x$
7. $1 \cdot 5x^{2} = 5x^{2}$
8. $1 \cdot 4x = 4x$
9. $1 \cdot 3 = 3$

Now combine all terms:

$S(x) = 15x^{4} + (12x^{3} + 10x^{3}) + (9x^{2} + 8x^{2} + 5x^{2}) + (6x + 4x) + 3$
$S(x) = 15x^{4} + 22x^{3} + 22x^{2} + 10x + 3$

Key Questions

• What is the result of adding $2x^{3} + 3x^{2} + 4$ and $5x^{3} + x + 1$?
• How do we multiply the polynomials $x + 2$ and $x^{2} - 3$?
• What is the degree of the polynomial $7x^{4} + 2x^{2} - x + 5$?
• Can you factor the polynomial $x^{2} - 9$?
• How would you evaluate the polynomial $P(x) = x^{2} + 3x + 2$ for $x = 2$?

Keywords

• Polynomial
• Addition
• Subtraction
• Multiplication
• Division