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The difference of the product of two binomials with the same terms is equal to the difference of the terms of the binomial.

Example: Which of the following is a special product?
a. [tex]x^2 - 36[/tex]
b. [tex]64y^2 - 100[/tex]
c. [tex]49x^2 + 25y^2[/tex]
d. [tex]144 - 36x^2y^2[/tex]
e. [tex]121a^2 - 144b^2c[/tex]

Sagot :

GinnyAnsweridn
To determine which of the given expressions is a special product, particularly focusing on the difference of squares, we need to identify expressions that can be written in the form [tex]\( a^2 - b^2 \)[/tex]. The difference of squares can be rewritten as:

[tex]\[ a^2 - b^2 = (a + b)(a - b) \][/tex]

Let's examine each of the given equations:

### Option a: [tex]\( x^2 - 36 \)[/tex]

This can be written as:

[tex]\[ x^2 - 6^2 \][/tex]

Here, [tex]\( a = x \)[/tex] and [tex]\( b = 6 \)[/tex]. This is indeed a difference of squares because it can be factored as:

[tex]\[ (x + 6)(x - 6) \][/tex]

### Option b: [tex]\( 64y^2 - 100 \)[/tex]

This can be written as:

[tex]\[ (8y)^2 - 10^2 \][/tex]

Here, [tex]\( a = 8y \)[/tex] and [tex]\( b = 10 \)[/tex]. This is also a difference of squares because it can be factored as:

[tex]\[ (8y + 10)(8y - 10) \][/tex]

### Option c: [tex]\( 49x^2 + 25y^2 \)[/tex]

This expression is a sum of squares and cannot be factored as a difference of squares. Therefore, it is not a special product of the type we are looking for.

### Option d: [tex]\( 144 - 36x^2y^2 \)[/tex]

This can be manipulated and written as:

[tex]\[ 12^2 - (6xy)^2 \][/tex]

Here, [tex]\( a = 12 \)[/tex] and [tex]\( b = 6xy \)[/tex]. This is a difference of squares because it can be factored as:

[tex]\[ (12 + 6xy)(12 - 6xy) \][/tex]

### Option e: [tex]\( 121a^2 - 144b^2c \)[/tex]

This cannot be rewritten in the form [tex]\( a^2 - b^2 \)[/tex] because the second term involves three different variables and can't be expressed as a perfect square of a single term.

### Conclusion:

Let's summarize the findings:

- Option a: [tex]\( x^2 - 36 \)[/tex] is a difference of squares.
- Option b: [tex]\( 64y^2 - 100 \)[/tex] is a difference of squares.
- Option c: [tex]\( 49x^2 + 25y^2 \)[/tex] is not a difference of squares.
- Option d: [tex]\( 144 - 36x^2y^2 \)[/tex] is a difference of squares.
- Option e: [tex]\( 121a^2 - 144b^2c \)[/tex] is not a difference of squares.

None of these options uniquely identify a single special product type (difference of squares) as each option check produces different results. Therefore, there is no correct answer among the given choices for the special product types based on the provided options and criteria.
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