Multiplication of Polynomials

Multiplying Every Term by Every Term

Similar to the operations of addition and subtraction of polynomials, the operation of multiplication on polynomials can also be understood through the basic concepts of number multiplication and the distributive property.

The main principle in multiplying two polynomials is: multiply each term in the first polynomial by each term in the second polynomial.

After performing all the multiplications between terms, the next step is to combine (add or subtract) like terms to simplify the result.

Multiplication Methods

There are several ways to perform polynomial multiplication, but all are based on the distributive property. The horizontal method writes every partial product in one line, and the table method places them in a grid before you combine like terms.

Horizontal Distribution Method

Multiply each term of the first polynomial by every term of the second polynomial, then combine like terms.

Example 1:

Find the product of $(x - 5)(x^2 + 3x - 1)$ .

(x - 5)(x^2 + 3x - 1)

Distribute $x-5$ :

=x(x^2+3x-1) -5(x^2+3x-1)

Calculate each term product:

\begin{aligned} &=(x\cdot x^2+x\cdot3x+x\cdot(-1)) \\ &\quad{}+(-5\cdot x^2-5\cdot3x-5\cdot(-1)) \end{aligned}

Calculate each term product:

\begin{aligned} &=(x^3+3x^2-x) \\ &\quad{}+(-5x^2-15x+5) \end{aligned}

Remove the parentheses:

=x^3+3x^2-x -5x^2-15x+5

Group like terms:

\begin{aligned} &=x^3+(3x^2-5x^2) \\ &\quad{}+(-x-15x)+5 \end{aligned}

The result is:

= x^3 - 2x^2 - 16x + 5

Example 2:

Find the product of $(x^2 - 2x + 7)(2x - 5)$ .

(x^2 - 2x + 7)(2x - 5)

Distribute $x^2-2x+7$ :

\begin{aligned} &=x^2(2x-5)-2x(2x-5) \\ &\quad{}+7(2x-5) \end{aligned}

Calculate each term product:

\begin{aligned} &=(x^2\cdot2x+x^2\cdot(-5)) \\ &\quad{}+(-2x\cdot2x-2x\cdot(-5)) \\ &\quad{}+(7\cdot2x+7\cdot(-5)) \end{aligned}

Calculate each term product:

\begin{aligned} &=(2x^3-5x^2)+(-4x^2+10x) \\ &\quad{}+(14x-35) \end{aligned}

Remove the parentheses:

=2x^3-5x^2-4x^2 +10x+14x-35

Group like terms:

\begin{aligned} &=2x^3+(-5x^2-4x^2) \\ &\quad{}+(10x+14x)-35 \end{aligned}

The result is:

= 2x^3 - 9x^2 + 24x - 35

Table Method with an Area Model

This method organizes the multiplication of each term using a table, similar to finding the area when multiplying two numbers.

For example, the multiplication $16 \times 12$ can be seen as the area of a rectangle with sides $10+6$ and $10+2$ .

	$10$	$6$
$10$	$100$	$60$
$2$	$20$	$12$

Total area is $100 + 60 + 20 + 12 = 192$ .

The same approach can be applied to polynomials.

Example 3:

Find the product of $(x + 6)(x + 2)$ using the table method.

	$x$	$+6$
$x$	$x^2$	$+6x$
$+2$	$+2x$	$+12$

Now, sum all the results inside the table cells:

x^2 + 6x + 2x + 12

Combine like terms:

= x^2 + (6x + 2x) + 12

Example 4:

Find the product of $(x - 5)(x^2 + 3x - 1)$ using the table method.

	$x^2$	$+3x$	$-1$
$x$	$x^3$	$+3x^2$	$-x$
$-5$	$-5x^2$	$-15x$	$+5$

Sum all the results inside the table cells:

x^3 + 3x^2 - x - 5x^2 - 15x + 5

Combine like terms:

\begin{aligned} &= x^3 + (3x^2 - 5x^2) \\ &\quad{}+ (-x - 15x) + 5 \end{aligned}

Both methods give the same result. The table method arranges the partial products so that no term is omitted.

Exercise

Expand and simplify $(2x-3)(x^2+x-4)$ . Distribute both terms of the first factor, then combine the terms with the same power of $x$ . A linear factor times a quadratic factor has degree at most three, so the result must contain no higher power than $x^3$ .

Worked Solution

Distribute both terms of the first factor across the second factor. Keep every partial product visible before combining like terms:

(2x-3)(x^2+x-4)

The check is structural: a linear factor times a quadratic factor can have degree at most $3$ , and the leading product $2x\cdot x^2=2x^3$ matches the leading term of the result.

Multiplying Every Term by Every Term

The main principle in multiplying two polynomials is: multiply each term in the first polynomial by each term in the second polynomial.

After performing all the multiplications between terms, the next step is to combine (add or subtract) like terms to simplify the result.

Multiplication Methods

Horizontal Distribution Method

Multiply each term of the first polynomial by every term of the second polynomial, then combine like terms.

Example 1:

Find the product of $(x - 5)(x^2 + 3x - 1)$ .

(x - 5)(x^2 + 3x - 1)

Distribute $x-5$ :

=x(x^2+3x-1) -5(x^2+3x-1)

Calculate each term product:

\begin{aligned} &=(x\cdot x^2+x\cdot3x+x\cdot(-1)) \\ &\quad{}+(-5\cdot x^2-5\cdot3x-5\cdot(-1)) \end{aligned}

Calculate each term product:

\begin{aligned} &=(x^3+3x^2-x) \\ &\quad{}+(-5x^2-15x+5) \end{aligned}

Remove the parentheses:

=x^3+3x^2-x -5x^2-15x+5

Group like terms:

\begin{aligned} &=x^3+(3x^2-5x^2) \\ &\quad{}+(-x-15x)+5 \end{aligned}

The result is:

= x^3 - 2x^2 - 16x + 5

Example 2:

Find the product of $(x^2 - 2x + 7)(2x - 5)$ .

(x^2 - 2x + 7)(2x - 5)

Distribute $x^2-2x+7$ :

\begin{aligned} &=x^2(2x-5)-2x(2x-5) \\ &\quad{}+7(2x-5) \end{aligned}

Calculate each term product:

\begin{aligned} &=(x^2\cdot2x+x^2\cdot(-5)) \\ &\quad{}+(-2x\cdot2x-2x\cdot(-5)) \\ &\quad{}+(7\cdot2x+7\cdot(-5)) \end{aligned}

Calculate each term product:

\begin{aligned} &=(2x^3-5x^2)+(-4x^2+10x) \\ &\quad{}+(14x-35) \end{aligned}

Remove the parentheses:

=2x^3-5x^2-4x^2 +10x+14x-35

Group like terms:

\begin{aligned} &=2x^3+(-5x^2-4x^2) \\ &\quad{}+(10x+14x)-35 \end{aligned}

The result is:

= 2x^3 - 9x^2 + 24x - 35

Table Method with an Area Model

This method organizes the multiplication of each term using a table, similar to finding the area when multiplying two numbers.

For example, the multiplication $16 \times 12$ can be seen as the area of a rectangle with sides $10+6$ and $10+2$ .

	$10$	$6$
$10$	$100$	$60$
$2$	$20$	$12$

Total area is $100 + 60 + 20 + 12 = 192$ .

The same approach can be applied to polynomials.

Example 3:

Find the product of $(x + 6)(x + 2)$ using the table method.

	$x$	$+6$
$x$	$x^2$	$+6x$
$+2$	$+2x$	$+12$

Now, sum all the results inside the table cells:

x^2 + 6x + 2x + 12

Combine like terms:

= x^2 + (6x + 2x) + 12

Example 4:

Find the product of $(x - 5)(x^2 + 3x - 1)$ using the table method.

	$x^2$	$+3x$	$-1$
$x$	$x^3$	$+3x^2$	$-x$
$-5$	$-5x^2$	$-15x$	$+5$

Sum all the results inside the table cells:

x^3 + 3x^2 - x - 5x^2 - 15x + 5

Combine like terms:

\begin{aligned} &= x^3 + (3x^2 - 5x^2) \\ &\quad{}+ (-x - 15x) + 5 \end{aligned}

Both methods give the same result. The table method arranges the partial products so that no term is omitted.

Exercise

Worked Solution

Distribute both terms of the first factor across the second factor. Keep every partial product visible before combining like terms:

(2x-3)(x^2+x-4)

The check is structural: a linear factor times a quadratic factor can have degree at most $3$ , and the leading product $2x\cdot x^2=2x^3$ matches the leading term of the result.