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Answer :
To factor the greatest common factor (GCF) from the polynomial
[tex]$$10x^6 + 15x^4 + 25x^3,$$[/tex]
follow these steps:
1. Determine the GCF of the coefficients:
The coefficients are 10, 15, and 25. The greatest common factor of these numbers is 5.
2. Determine the GCF of the variable part:
The variable [tex]$x$[/tex] has exponents 6, 4, and 3 in the three terms. The smallest exponent is 3. Thus, the GCF for the variable is [tex]$x^3$[/tex].
3. Combine the results:
The overall GCF for the polynomial is [tex]$5x^3$[/tex].
4. Factor out the GCF from each term:
- For the first term, [tex]$$\frac{10x^6}{5x^3} = 2x^{6-3} = 2x^3.$$[/tex]
- For the second term, [tex]$$\frac{15x^4}{5x^3} = 3x^{4-3} = 3x.$$[/tex]
- For the third term, [tex]$$\frac{25x^3}{5x^3} = 5x^{3-3} = 5.$$[/tex]
5. Write the fully factored expression:
Factor out the [tex]$5x^3$[/tex] to obtain:
[tex]$$5x^3(2x^3 + 3x + 5).$$[/tex]
Thus, the final answer is:
[tex]$$5x^3(2x^3+3x+5).$$[/tex]
[tex]$$10x^6 + 15x^4 + 25x^3,$$[/tex]
follow these steps:
1. Determine the GCF of the coefficients:
The coefficients are 10, 15, and 25. The greatest common factor of these numbers is 5.
2. Determine the GCF of the variable part:
The variable [tex]$x$[/tex] has exponents 6, 4, and 3 in the three terms. The smallest exponent is 3. Thus, the GCF for the variable is [tex]$x^3$[/tex].
3. Combine the results:
The overall GCF for the polynomial is [tex]$5x^3$[/tex].
4. Factor out the GCF from each term:
- For the first term, [tex]$$\frac{10x^6}{5x^3} = 2x^{6-3} = 2x^3.$$[/tex]
- For the second term, [tex]$$\frac{15x^4}{5x^3} = 3x^{4-3} = 3x.$$[/tex]
- For the third term, [tex]$$\frac{25x^3}{5x^3} = 5x^{3-3} = 5.$$[/tex]
5. Write the fully factored expression:
Factor out the [tex]$5x^3$[/tex] to obtain:
[tex]$$5x^3(2x^3 + 3x + 5).$$[/tex]
Thus, the final answer is:
[tex]$$5x^3(2x^3+3x+5).$$[/tex]
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