Multiplication of Polynomials: Polynomial - Mathematics (Grade 11)

Basic Principle of Polynomial Multiplication

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.

Horizontal Distribution Method

This method involves distributing each term of the first polynomial to all terms of the second polynomial.

Example 1:

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

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

= x(x^2 + 3x - 1) - 5(x^2 + 3x - 1) \quad \text{(Distribute } (x-5))

= (x \cdot x^2 + x \cdot 3x + x \cdot (-1)) + (-5 \cdot x^2 - 5 \cdot 3x - 5 \cdot (-1)) \quad \text{(Distribute } x \text{ and } -5)

= (x^3 + 3x^2 - x) + (-5x^2 - 15x + 5) \quad \text{(Result of term multiplication)}

= x^3 + 3x^2 - x - 5x^2 - 15x + 5 \quad \text{(Remove parentheses)}

= x^3 + (3x^2 - 5x^2) + (-x - 15x) + 5 \quad \text{(Group like terms)}

= x^3 - 2x^2 - 16x + 5 \quad \text{(Final result)}

Example 2:

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

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

= x^2(2x - 5) - 2x(2x - 5) + 7(2x - 5) \quad \text{(Distribute } (x^2-2x+7))

= (x^2 \cdot 2x + x^2 \cdot (-5)) + (-2x \cdot 2x - 2x \cdot (-5)) + (7 \cdot 2x + 7 \cdot (-5)) \quad \text{(Distribute } x^2, -2x, \text{ and } 7)

= (2x^3 - 5x^2) + (-4x^2 + 10x) + (14x - 35) \quad \text{(Result of term multiplication)}

= 2x^3 - 5x^2 - 4x^2 + 10x + 14x - 35 \quad \text{(Remove parentheses)}

= 2x^3 + (-5x^2 - 4x^2) + (10x + 14x) - 35 \quad \text{(Group like terms)}

= 2x^3 - 9x^2 + 24x - 35 \quad \text{(Final result)}

Table Method (Area Model Analogy)

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 = $100 + 60 + 20 + 12 = 192$ .

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

= x^2 + 8x + 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:

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

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

Notice that the result from the table method is the same as the result from the horizontal distribution method. The table method is just another way to organize the multiplication of each term.

Basic Principle of Polynomial Multiplication

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.

Horizontal Distribution Method

This method involves distributing each term of the first polynomial to all terms of the second polynomial.

Example 1:

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

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

= x(x^2 + 3x - 1) - 5(x^2 + 3x - 1) \quad \text{(Distribute } (x-5))

= (x \cdot x^2 + x \cdot 3x + x \cdot (-1)) + (-5 \cdot x^2 - 5 \cdot 3x - 5 \cdot (-1)) \quad \text{(Distribute } x \text{ and } -5)

= (x^3 + 3x^2 - x) + (-5x^2 - 15x + 5) \quad \text{(Result of term multiplication)}

= x^3 + 3x^2 - x - 5x^2 - 15x + 5 \quad \text{(Remove parentheses)}

= x^3 + (3x^2 - 5x^2) + (-x - 15x) + 5 \quad \text{(Group like terms)}

= x^3 - 2x^2 - 16x + 5 \quad \text{(Final result)}

Example 2:

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

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

= x^2(2x - 5) - 2x(2x - 5) + 7(2x - 5) \quad \text{(Distribute } (x^2-2x+7))

= (x^2 \cdot 2x + x^2 \cdot (-5)) + (-2x \cdot 2x - 2x \cdot (-5)) + (7 \cdot 2x + 7 \cdot (-5)) \quad \text{(Distribute } x^2, -2x, \text{ and } 7)

= (2x^3 - 5x^2) + (-4x^2 + 10x) + (14x - 35) \quad \text{(Result of term multiplication)}

= 2x^3 - 5x^2 - 4x^2 + 10x + 14x - 35 \quad \text{(Remove parentheses)}

= 2x^3 + (-5x^2 - 4x^2) + (10x + 14x) - 35 \quad \text{(Group like terms)}

= 2x^3 - 9x^2 + 24x - 35 \quad \text{(Final result)}

Table Method (Area Model Analogy)

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 = $100 + 60 + 20 + 12 = 192$ .

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

= x^2 + 8x + 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:

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

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

Notice that the result from the table method is the same as the result from the horizontal distribution method. The table method is just another way to organize the multiplication of each term.

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