Answer :
Certainly! Let's match each polynomial with the appropriate factoring pattern. Here's a detailed explanation of each:
1. [tex]\(25x^2 - 16\)[/tex]:
- This expression is in the form of [tex]\(a^2 - b^2\)[/tex], which is a classic Difference of Squares. It can be factored as [tex]\((5x)^2 - (4)^2\)[/tex]. The formula for factoring is [tex]\(a^2 - b^2 = (a + b)(a - b)\)[/tex].
2. [tex]\(x^3 - 125\)[/tex]:
- This polynomial is a difference of cubes since both terms are perfect cubes: [tex]\(x^3\)[/tex] and [tex]\(5^3\)[/tex]. The formula for the Difference of Cubes is [tex]\(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)[/tex]. In this case, it's [tex]\((x - 5)(x^2 + 5x + 25)\)[/tex].
3. [tex]\(8x^3 + 27y^3\)[/tex]:
- This is a sum of cubes, where the terms [tex]\(2x\)[/tex] and [tex]\(3y\)[/tex] are cubed, i.e., [tex]\((2x)^3 + (3y)^3\)[/tex]. The formula for the Sum of Cubes is [tex]\(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)[/tex]. For this polynomial, it's [tex]\((2x + 3y)((2x)^2 - 2x(3y) + (3y)^2)\)[/tex].
4. [tex]\(25x^2 - 20xy + 4y^2\)[/tex]:
- This can be recognized as a Perfect Square Trinomial since it is of the form [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex]. It can be written as [tex]\((5x - 2y)^2\)[/tex].
5. [tex]\(25x^3 - 8xy^2\)[/tex]:
- Here, we can look for a Greatest Common Factor. Both terms share a factor of [tex]\(x\)[/tex], giving us [tex]\(x(25x^2 - 8y^2)\)[/tex].
In summary, here are the matches:
- [tex]\(25x^2 - 16\)[/tex]: Difference of Squares
- [tex]\(x^3 - 125\)[/tex]: Difference of Cubes
- [tex]\(8x^3 + 27y^3\)[/tex]: Sum of Cubes
- [tex]\(25x^2 - 20xy + 4y^2\)[/tex]: Perfect Square Trinomial
- [tex]\(25x^3 - 8xy^2\)[/tex]: Greatest Common Factor
This is how each polynomial aligns with its respective factoring pattern.
1. [tex]\(25x^2 - 16\)[/tex]:
- This expression is in the form of [tex]\(a^2 - b^2\)[/tex], which is a classic Difference of Squares. It can be factored as [tex]\((5x)^2 - (4)^2\)[/tex]. The formula for factoring is [tex]\(a^2 - b^2 = (a + b)(a - b)\)[/tex].
2. [tex]\(x^3 - 125\)[/tex]:
- This polynomial is a difference of cubes since both terms are perfect cubes: [tex]\(x^3\)[/tex] and [tex]\(5^3\)[/tex]. The formula for the Difference of Cubes is [tex]\(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)[/tex]. In this case, it's [tex]\((x - 5)(x^2 + 5x + 25)\)[/tex].
3. [tex]\(8x^3 + 27y^3\)[/tex]:
- This is a sum of cubes, where the terms [tex]\(2x\)[/tex] and [tex]\(3y\)[/tex] are cubed, i.e., [tex]\((2x)^3 + (3y)^3\)[/tex]. The formula for the Sum of Cubes is [tex]\(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)[/tex]. For this polynomial, it's [tex]\((2x + 3y)((2x)^2 - 2x(3y) + (3y)^2)\)[/tex].
4. [tex]\(25x^2 - 20xy + 4y^2\)[/tex]:
- This can be recognized as a Perfect Square Trinomial since it is of the form [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex]. It can be written as [tex]\((5x - 2y)^2\)[/tex].
5. [tex]\(25x^3 - 8xy^2\)[/tex]:
- Here, we can look for a Greatest Common Factor. Both terms share a factor of [tex]\(x\)[/tex], giving us [tex]\(x(25x^2 - 8y^2)\)[/tex].
In summary, here are the matches:
- [tex]\(25x^2 - 16\)[/tex]: Difference of Squares
- [tex]\(x^3 - 125\)[/tex]: Difference of Cubes
- [tex]\(8x^3 + 27y^3\)[/tex]: Sum of Cubes
- [tex]\(25x^2 - 20xy + 4y^2\)[/tex]: Perfect Square Trinomial
- [tex]\(25x^3 - 8xy^2\)[/tex]: Greatest Common Factor
This is how each polynomial aligns with its respective factoring pattern.
