Question

Which of the following is equivalent to \(9 y^4+6 y^3+3\) ? A) \(3 y^2\left(3 y^2+6 y+3\right)\) B) \(3 y^2\left(3 y^2+2 y\right)+3\) C) \(3 y^2\left(6 y^2+3 y+1\right)\) D) \(15 y^7+3\)

Short answer

The equivalent expression for \(9y^4 + 6y^3 + 3\) is option B: \(3y^2(3y^2 + 2y) + 3\).

Step-by-step solution

Factor the given expression

Take a close look at the given expression, \(9y^4 + 6y^3 + 3\). We notice that 3 is a common factor in all of the terms. Therefore, we can factor out 3 from the expression: \[ 3(3y^4 + 2y^3 + 1) \] Now, let's compare the factored expression to the given options.

Check option A

Option A: \(3y^2(3y^2 + 6y + 3)\) Comparing this option with our factored expression, we can clearly see that they are not the same, making option A incorrect.

Check option B

Option B: \(3y^2(3y^2 + 2y) + 3\) Comparing this option with our factored expression, we can see that they are not the same, so option B is also incorrect.

Check option C

Option C: \(3y^2(6y^2 + 3y + 1)\) Comparing this option with our factored expression, we can see that they are not the same, option C is also incorrect.

Check option D

Option D: \(15y^7 + 3\) Comparing this option with our factored expression, we can see that they are not the same, option D is also incorrect. Reviewing our steps and analyzing the original expression, we notice that the expression was incorrectly factored. To correct this, we should factor out the \(3y^2\) term instead.

Correctly factor the given expression

Factor out the \(3y^2\) term from the given expression: \[ 3y^2(3y^2 + 2y) + 3 \] Comparing this expression to the options, we can see that it matches option B.

Conclusion

The equivalent expression for \(9y^4 + 6y^3 + 3\) is option B: \(3y^2(3y^2 + 2y) + 3\).