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Answer :
To factor the polynomial [tex]\( x^3 + 3x^2 + 16x + 48 \)[/tex] completely, follow these steps:
1. Look for common factors: Initially, check if there's a common factor in all terms. In this case, there isn't any common factor other than 1.
2. Factor by grouping: The polynomial has four terms, which suggests you might be able to use factoring by grouping. Group the first two terms and the last two terms:
[tex]\[
(x^3 + 3x^2) + (16x + 48)
\][/tex]
3. Factor each group:
- For the first group [tex]\( x^3 + 3x^2 \)[/tex], factor out the greatest common factor, which is [tex]\( x^2 \)[/tex]:
[tex]\[
x^2(x + 3)
\][/tex]
- For the second group [tex]\( 16x + 48 \)[/tex], factor out the greatest common factor, which is 16:
[tex]\[
16(x + 3)
\][/tex]
4. Combine the factored groups: You can now notice that [tex]\( (x + 3) \)[/tex] is a common factor:
[tex]\[
x^2(x + 3) + 16(x + 3) = (x^2 + 16)(x + 3)
\][/tex]
5. Check if further factoring is possible:
- The quadratic [tex]\( x^2 + 16 \)[/tex] cannot be factored further over the integers since it does not break down into factors with integer coefficients.
So, the completely factored form of the polynomial [tex]\( x^3 + 3x^2 + 16x + 48 \)[/tex] is:
[tex]\[
(x + 3)(x^2 + 16)
\][/tex]
1. Look for common factors: Initially, check if there's a common factor in all terms. In this case, there isn't any common factor other than 1.
2. Factor by grouping: The polynomial has four terms, which suggests you might be able to use factoring by grouping. Group the first two terms and the last two terms:
[tex]\[
(x^3 + 3x^2) + (16x + 48)
\][/tex]
3. Factor each group:
- For the first group [tex]\( x^3 + 3x^2 \)[/tex], factor out the greatest common factor, which is [tex]\( x^2 \)[/tex]:
[tex]\[
x^2(x + 3)
\][/tex]
- For the second group [tex]\( 16x + 48 \)[/tex], factor out the greatest common factor, which is 16:
[tex]\[
16(x + 3)
\][/tex]
4. Combine the factored groups: You can now notice that [tex]\( (x + 3) \)[/tex] is a common factor:
[tex]\[
x^2(x + 3) + 16(x + 3) = (x^2 + 16)(x + 3)
\][/tex]
5. Check if further factoring is possible:
- The quadratic [tex]\( x^2 + 16 \)[/tex] cannot be factored further over the integers since it does not break down into factors with integer coefficients.
So, the completely factored form of the polynomial [tex]\( x^3 + 3x^2 + 16x + 48 \)[/tex] is:
[tex]\[
(x + 3)(x^2 + 16)
\][/tex]
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