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Answer :
To factor the polynomial [tex]\( 45x^3 - 15x^2 + 48x - 16 \)[/tex] completely, we follow these steps:
1. Group and Factor: Start by grouping the terms in pairs and factor out the common factor in each group.
- The polynomial can be rewritten as:
[tex]\[
(45x^3 - 15x^2) + (48x - 16)
\][/tex]
- For the first group [tex]\( (45x^3 - 15x^2) \)[/tex], factor out the greatest common factor, which is [tex]\( 15x^2 \)[/tex]:
[tex]\[
15x^2(3x - 1)
\][/tex]
- For the second group [tex]\( (48x - 16) \)[/tex], factor out the greatest common factor, which is [tex]\( 16 \)[/tex]:
[tex]\[
16(3x - 1)
\][/tex]
2. Combine the Factorization: Notice that both groups contain a common factor of [tex]\( (3x - 1) \)[/tex].
[tex]\[
15x^2(3x - 1) + 16(3x - 1) = (3x - 1)(15x^2 + 16)
\][/tex]
3. Check for Further Factorization: Now, let's verify if the quadratic [tex]\( 15x^2 + 16 \)[/tex] can be factored further.
- Check if the quadratic expression can be factored using standard methods like finding roots or using the quadratic formula. In this case, the expression [tex]\( 15x^2 + 16 \)[/tex] does not factor further over the integers.
4. Final Answer: Thus, the completely factored form of the polynomial is:
[tex]\[
(3x - 1)(15x^2 + 16)
\][/tex]
This is your complete factorization. If any further factoring is required or possible, it would be outside standard integer factorization.
1. Group and Factor: Start by grouping the terms in pairs and factor out the common factor in each group.
- The polynomial can be rewritten as:
[tex]\[
(45x^3 - 15x^2) + (48x - 16)
\][/tex]
- For the first group [tex]\( (45x^3 - 15x^2) \)[/tex], factor out the greatest common factor, which is [tex]\( 15x^2 \)[/tex]:
[tex]\[
15x^2(3x - 1)
\][/tex]
- For the second group [tex]\( (48x - 16) \)[/tex], factor out the greatest common factor, which is [tex]\( 16 \)[/tex]:
[tex]\[
16(3x - 1)
\][/tex]
2. Combine the Factorization: Notice that both groups contain a common factor of [tex]\( (3x - 1) \)[/tex].
[tex]\[
15x^2(3x - 1) + 16(3x - 1) = (3x - 1)(15x^2 + 16)
\][/tex]
3. Check for Further Factorization: Now, let's verify if the quadratic [tex]\( 15x^2 + 16 \)[/tex] can be factored further.
- Check if the quadratic expression can be factored using standard methods like finding roots or using the quadratic formula. In this case, the expression [tex]\( 15x^2 + 16 \)[/tex] does not factor further over the integers.
4. Final Answer: Thus, the completely factored form of the polynomial is:
[tex]\[
(3x - 1)(15x^2 + 16)
\][/tex]
This is your complete factorization. If any further factoring is required or possible, it would be outside standard integer factorization.
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