8.5 - Multiplying a Polynomial by a Polynomial

In this section, we learn how to multiply two polynomials. A polynomial is an expression consisting of variables and coefficients, involving operations of addition, subtraction, and multiplication, raised to non-negative integer powers. When multiplying polynomials, we apply the distributive property and expand the product step by step.

Let us consider two polynomials, $P(x)$ and $Q(x)$, such that:

• $P(x) = a_1x^n + a_2x^{n-1} + \dots + a_n$
• $Q(x) = b_1x^m + b_2x^{m-1} + \dots + b_m$

Multiplying $P(x)$ and $Q(x)$ involves distributing each term of $P(x)$ to each term of $Q(x)$. This method is also known as the distributive property or the "FOIL" method when dealing with binomials.

The distributive property states that:

$$ (a + b)(c + d) = ac + ad + bc + bd $$

When multiplying two polynomials, we apply this idea across all terms in each polynomial.

Step-by-Step Multiplication

Let’s take an example to illustrate the process. Consider the following polynomials:

$$ P(x) = (2x + 3) \quad \text{and} \quad Q(x) = (x + 4) $$

We will multiply these two polynomials using the distributive property:

$$ (2x + 3)(x + 4) $$

First, distribute $2x$ to both terms in $(x + 4)$:

$$ 2x \cdot x = 2x^2 $$ $$ 2x \cdot 4 = 8x $$

Next, distribute $3$ to both terms in $(x + 4)$:

$$ 3 \cdot x = 3x $$ $$ 3 \cdot 4 = 12 $$

Now, combine all the terms:

$$ 2x^2 + 8x + 3x + 12 $$

Finally, combine like terms:

$$ 2x^2 + 11x + 12 $$

This is the product of the two polynomials. So, $P(x) \cdot Q(x) = 2x^2 + 11x + 12$.

Multiplying Polynomials with More Terms

For polynomials with more terms, follow the same approach. For example, if we multiply:

$$ P(x) = (x^2 + 2x + 1) \quad \text{and} \quad Q(x) = (x + 3) $$

We distribute each term of $P(x)$ to every term in $Q(x)$:

$$ (x^2 + 2x + 1)(x + 3) $$

Distribute $x^2$:

$$ x^2 \cdot x = x^3 $$ $$ x^2 \cdot 3 = 3x^2 $$

Distribute $2x$:

$$ 2x \cdot x = 2x^2 $$ $$ 2x \cdot 3 = 6x $$

Distribute $1$:

$$ 1 \cdot x = x $$ $$ 1 \cdot 3 = 3 $$

Now, combine all terms:

$$ x^3 + 3x^2 + 2x^2 + 6x + x + 3 $$

Combine like terms:

$$ x^3 + 5x^2 + 7x + 3 $$

Thus, the product of the polynomials is:

$$ P(x) \cdot Q(x) = x^3 + 5x^2 + 7x + 3 $$

Key Points to Remember

• Each term in the first polynomial must be multiplied by each term in the second polynomial.
• Always combine like terms after performing the multiplication.
• The distributive property is the fundamental rule to follow while multiplying polynomials.

Example Problems

1. Multiply the following polynomials:

• $(x + 2)(x + 5)$
• $(2x - 3)(x + 1)$

2. Multiply and simplify the expression:

• $(x^2 + 3x + 2)(x + 4)$