At Westonci.ca, we connect you with the best answers from a community of experienced and knowledgeable individuals. Get quick and reliable solutions to your questions from a community of seasoned experts on our user-friendly platform. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
Sagot :
Sure, let's factorize the polynomial [tex]\(12x^2 + 36x + 27\)[/tex] step by step to find the correct option among the given choices.
### Step 1: Identify Common Factors
First we look for common factors in the polynomial [tex]\(12x^2 + 36x + 27\)[/tex].
The greatest common divisor (GCD) of the coefficients 12, 36, and 27 is 3. We can factor out this common factor:
[tex]\[ 12x^2 + 36x + 27 = 3(4x^2 + 12x + 9) \][/tex]
### Step 2: Factor the Quadratic Inside
Next, we need to factor the quadratic expression [tex]\(4x^2 + 12x + 9\)[/tex].
To factor this trinomial, we look for two numbers that multiply to [tex]\(4 \times 9 = 36\)[/tex] and add to 12. These numbers are 6 and 6. Thus, we can rewrite the quadratic:
[tex]\[ 4x^2 + 12x + 9 = 4x^2 + 6x + 6x + 9 \][/tex]
### Step 3: Factor by Grouping
Now we can factor by grouping:
[tex]\[ 4x^2 + 6x + 6x + 9 = (4x^2 + 6x) + (6x + 9) \][/tex]
Factor out the common terms in each group:
[tex]\[ = 2x(2x + 3) + 3(2x + 3) \][/tex]
Notice that [tex]\((2x + 3)\)[/tex] is a common factor:
[tex]\[ = (2x + 3)(2x + 3) \][/tex]
This can be written as:
[tex]\[ = (2x + 3)^2 \][/tex]
### Step 4: Incorporate the Common Factor
Recall we factored out a 3 initially. Now we combine it back:
[tex]\[ 12x^2 + 36x + 27 = 3(4x^2 + 12x + 9) = 3(2x + 3)^2 \][/tex]
### Final Answer
Now we compare our result with the given options:
A. [tex]\((x + 3)(12x + 9)\)[/tex]
B. [tex]\(3(2x + 3)^2\)[/tex]
C. [tex]\(6(x + 2)^2\)[/tex]
D. [tex]\((6x + 3)(2x + 9)\)[/tex]
The correct factorization is:
[tex]\[ 12 x^2 + 36 x + 27 = 3(2 x + 3)^2 \][/tex]
This matches option B.
### Conclusion
The complete factorization of [tex]\(12 x^2+36 x+27\)[/tex] is given by:
[tex]\[ \boxed{B. \, 3(2 x + 3)^2} \][/tex]
### Step 1: Identify Common Factors
First we look for common factors in the polynomial [tex]\(12x^2 + 36x + 27\)[/tex].
The greatest common divisor (GCD) of the coefficients 12, 36, and 27 is 3. We can factor out this common factor:
[tex]\[ 12x^2 + 36x + 27 = 3(4x^2 + 12x + 9) \][/tex]
### Step 2: Factor the Quadratic Inside
Next, we need to factor the quadratic expression [tex]\(4x^2 + 12x + 9\)[/tex].
To factor this trinomial, we look for two numbers that multiply to [tex]\(4 \times 9 = 36\)[/tex] and add to 12. These numbers are 6 and 6. Thus, we can rewrite the quadratic:
[tex]\[ 4x^2 + 12x + 9 = 4x^2 + 6x + 6x + 9 \][/tex]
### Step 3: Factor by Grouping
Now we can factor by grouping:
[tex]\[ 4x^2 + 6x + 6x + 9 = (4x^2 + 6x) + (6x + 9) \][/tex]
Factor out the common terms in each group:
[tex]\[ = 2x(2x + 3) + 3(2x + 3) \][/tex]
Notice that [tex]\((2x + 3)\)[/tex] is a common factor:
[tex]\[ = (2x + 3)(2x + 3) \][/tex]
This can be written as:
[tex]\[ = (2x + 3)^2 \][/tex]
### Step 4: Incorporate the Common Factor
Recall we factored out a 3 initially. Now we combine it back:
[tex]\[ 12x^2 + 36x + 27 = 3(4x^2 + 12x + 9) = 3(2x + 3)^2 \][/tex]
### Final Answer
Now we compare our result with the given options:
A. [tex]\((x + 3)(12x + 9)\)[/tex]
B. [tex]\(3(2x + 3)^2\)[/tex]
C. [tex]\(6(x + 2)^2\)[/tex]
D. [tex]\((6x + 3)(2x + 9)\)[/tex]
The correct factorization is:
[tex]\[ 12 x^2 + 36 x + 27 = 3(2 x + 3)^2 \][/tex]
This matches option B.
### Conclusion
The complete factorization of [tex]\(12 x^2+36 x+27\)[/tex] is given by:
[tex]\[ \boxed{B. \, 3(2 x + 3)^2} \][/tex]
Thanks for using our service. We aim to provide the most accurate answers for all your queries. Visit us again for more insights. Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.