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Sagot :
Sure, let's factor the given polynomials one by one.
### 1. Polynomial: [tex]\( 7c^2 - 25 \)[/tex]
This polynomial is of the form [tex]\( a^2 - b^2 \)[/tex], which is a difference of squares. The difference of squares formula is:
[tex]\[ a^2 - b^2 = (a + b)(a - b) \][/tex]
However, in this case, we must check if the given polynomial fits this formula perfectly. [tex]\( 7c^2 - 25 \)[/tex] does not represent a perfect square difference straightforwardly.
Upon inspection, 7 is not a perfect square, and thus this polynomial is already factored as much as possible:
[tex]\[ 7c^2 - 25 \][/tex]
### 2. Polynomial: [tex]\( 4k^4 - 100 \)[/tex]
This polynomial also resembles the difference of squares:
[tex]\[ a^2 - b^2 = (a + b)(a - b) \][/tex]
First, express the polynomial as:
[tex]\[ 4k^4 - 100 = (2k^2)^2 - (10)^2 \][/tex]
Applying the difference of squares:
[tex]\[ (2k^2 + 10)(2k^2 - 10) \][/tex]
Then, notice that [tex]\( 2k^2 - 10 \)[/tex] can be further factored as:
[tex]\[ 2(k^2 - 5) \][/tex]
Thus, the complete factorization is:
[tex]\[ (2k^2 + 10)(2(k^2 - 5)) \][/tex]
We notice that there is a common factor of 2:
[tex]\[ 2(2k^2 + 10)(k^2 - 5) \][/tex]
### 3. Polynomial: [tex]\( 2w^4 - 1 \)[/tex]
Expressing this polynomial as:
[tex]\[ 2w^4 - 1 = (w^2)^2 - \left(\frac{1}{\sqrt{2}}\right)^2 \][/tex]
This is similar to a difference of squares, but we must handle irrational numbers. Thus the factorization is non-trivial:
[tex]\[ (2w^2 - 1)(w^2 + \frac{1}{\sqrt{2}}) \][/tex]
### 4. Polynomial: [tex]\( 1 - r^{10} \)[/tex]
This polynomial can be expressed as:
[tex]\[ 1 - r^{10} = (1)^2 - (r^5)^2 \][/tex]
Applying the difference of squares we get:
[tex]\[ (1 + r^5)(1 - r^5) \][/tex]
### 5. Polynomial: [tex]\( 336 - h^6 \)[/tex]
This polynomial can be expressed as:
[tex]\[ 336 - h^6\][/tex]
This does not straightly fit into a perfect square difference.
Thus, the factor already represents all factors.
Thus, the completely factored forms are:
1. [tex]\( 7c^2 - 25 \)[/tex] (already factored)
2. [tex]\( 2(2k^2 + 10)(k^2 - 5) \)[/tex]
3. [tex]\( 2w^2 - 1 \)[/tex]
4. [tex]\( (1 + r^5)(1 - r^5) \)[/tex]
5. [tex]\( 336 - h^6 \)[/tex]
### 1. Polynomial: [tex]\( 7c^2 - 25 \)[/tex]
This polynomial is of the form [tex]\( a^2 - b^2 \)[/tex], which is a difference of squares. The difference of squares formula is:
[tex]\[ a^2 - b^2 = (a + b)(a - b) \][/tex]
However, in this case, we must check if the given polynomial fits this formula perfectly. [tex]\( 7c^2 - 25 \)[/tex] does not represent a perfect square difference straightforwardly.
Upon inspection, 7 is not a perfect square, and thus this polynomial is already factored as much as possible:
[tex]\[ 7c^2 - 25 \][/tex]
### 2. Polynomial: [tex]\( 4k^4 - 100 \)[/tex]
This polynomial also resembles the difference of squares:
[tex]\[ a^2 - b^2 = (a + b)(a - b) \][/tex]
First, express the polynomial as:
[tex]\[ 4k^4 - 100 = (2k^2)^2 - (10)^2 \][/tex]
Applying the difference of squares:
[tex]\[ (2k^2 + 10)(2k^2 - 10) \][/tex]
Then, notice that [tex]\( 2k^2 - 10 \)[/tex] can be further factored as:
[tex]\[ 2(k^2 - 5) \][/tex]
Thus, the complete factorization is:
[tex]\[ (2k^2 + 10)(2(k^2 - 5)) \][/tex]
We notice that there is a common factor of 2:
[tex]\[ 2(2k^2 + 10)(k^2 - 5) \][/tex]
### 3. Polynomial: [tex]\( 2w^4 - 1 \)[/tex]
Expressing this polynomial as:
[tex]\[ 2w^4 - 1 = (w^2)^2 - \left(\frac{1}{\sqrt{2}}\right)^2 \][/tex]
This is similar to a difference of squares, but we must handle irrational numbers. Thus the factorization is non-trivial:
[tex]\[ (2w^2 - 1)(w^2 + \frac{1}{\sqrt{2}}) \][/tex]
### 4. Polynomial: [tex]\( 1 - r^{10} \)[/tex]
This polynomial can be expressed as:
[tex]\[ 1 - r^{10} = (1)^2 - (r^5)^2 \][/tex]
Applying the difference of squares we get:
[tex]\[ (1 + r^5)(1 - r^5) \][/tex]
### 5. Polynomial: [tex]\( 336 - h^6 \)[/tex]
This polynomial can be expressed as:
[tex]\[ 336 - h^6\][/tex]
This does not straightly fit into a perfect square difference.
Thus, the factor already represents all factors.
Thus, the completely factored forms are:
1. [tex]\( 7c^2 - 25 \)[/tex] (already factored)
2. [tex]\( 2(2k^2 + 10)(k^2 - 5) \)[/tex]
3. [tex]\( 2w^2 - 1 \)[/tex]
4. [tex]\( (1 + r^5)(1 - r^5) \)[/tex]
5. [tex]\( 336 - h^6 \)[/tex]
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