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Answer :
To factor the trinomial [tex]\( 6x^3 + 27x^2 + 78x \)[/tex] completely, follow these steps:
1. Identify the Greatest Common Factor (GCF):
First, look for the GCF of the coefficients and the variable terms. The coefficients are 6, 27, and 78, and all terms contain [tex]\( x \)[/tex].
- The GCF of 6, 27, and 78 is 3 because each number is divisible by 3.
- The variable part [tex]\( x \)[/tex] is common to all terms and the smallest power of [tex]\( x \)[/tex] is [tex]\( x \)[/tex].
Therefore, the GCF is [tex]\( 3x \)[/tex].
2. Factor out the GCF:
Divide each term by the GCF, [tex]\( 3x \)[/tex]:
[tex]\[
6x^3 \div 3x = 2x^2
\][/tex]
[tex]\[
27x^2 \div 3x = 9x
\][/tex]
[tex]\[
78x \div 3x = 26
\][/tex]
Now, factor [tex]\( 3x \)[/tex] out of the trinomial:
[tex]\[
6x^3 + 27x^2 + 78x = 3x(2x^2 + 9x + 26)
\][/tex]
3. Check for Further Factorization:
Next, we check if the quadratic [tex]\( 2x^2 + 9x + 26 \)[/tex] can be factored further.
We look for two numbers that multiply to give [tex]\( 2 \times 26 = 52 \)[/tex] and add to give [tex]\( 9 \)[/tex].
No such integer pairs exist, suggesting that [tex]\( 2x^2 + 9x + 26 \)[/tex] is not factorable over the integers.
Since we cannot factor the quadratic expression further, we conclude that the trinomial is factored completely as:
[tex]\[ 6x^3 + 27x^2 + 78x = 3x(2x^2 + 9x + 26) \][/tex]
Therefore, the correct choice is:
[tex]\[ \boxed{3x(2x^2 + 9x + 26)} \][/tex]
1. Identify the Greatest Common Factor (GCF):
First, look for the GCF of the coefficients and the variable terms. The coefficients are 6, 27, and 78, and all terms contain [tex]\( x \)[/tex].
- The GCF of 6, 27, and 78 is 3 because each number is divisible by 3.
- The variable part [tex]\( x \)[/tex] is common to all terms and the smallest power of [tex]\( x \)[/tex] is [tex]\( x \)[/tex].
Therefore, the GCF is [tex]\( 3x \)[/tex].
2. Factor out the GCF:
Divide each term by the GCF, [tex]\( 3x \)[/tex]:
[tex]\[
6x^3 \div 3x = 2x^2
\][/tex]
[tex]\[
27x^2 \div 3x = 9x
\][/tex]
[tex]\[
78x \div 3x = 26
\][/tex]
Now, factor [tex]\( 3x \)[/tex] out of the trinomial:
[tex]\[
6x^3 + 27x^2 + 78x = 3x(2x^2 + 9x + 26)
\][/tex]
3. Check for Further Factorization:
Next, we check if the quadratic [tex]\( 2x^2 + 9x + 26 \)[/tex] can be factored further.
We look for two numbers that multiply to give [tex]\( 2 \times 26 = 52 \)[/tex] and add to give [tex]\( 9 \)[/tex].
No such integer pairs exist, suggesting that [tex]\( 2x^2 + 9x + 26 \)[/tex] is not factorable over the integers.
Since we cannot factor the quadratic expression further, we conclude that the trinomial is factored completely as:
[tex]\[ 6x^3 + 27x^2 + 78x = 3x(2x^2 + 9x + 26) \][/tex]
Therefore, the correct choice is:
[tex]\[ \boxed{3x(2x^2 + 9x + 26)} \][/tex]
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