Question
1) \( \left(4 v^{4}+16 v^{3}+16 v^{2}\right) \div 4 v^{2} \)
Ask by Gonzalez Stewart. in the United States
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The simplified form of the expression is \( v^{2} + 4v + 4 \) or \( (v + 2)^2 \).
Solution
To simplify the expression:
\[
\frac{4v^{4} + 16v^{3} + 16v^{2}}{4v^{2}}
\]
**Step 1: Factor out the common term in the numerator**
Notice that each term in the numerator has a common factor of \(4v^{2}\):
\[
4v^{4} + 16v^{3} + 16v^{2} = 4v^{2}(v^{2} + 4v + 4)
\]
**Step 2: Divide the factored numerator by the denominator**
\[
\frac{4v^{2}(v^{2} + 4v + 4)}{4v^{2}}
\]
The \(4v^{2}\) in the numerator and the denominator cancel out:
\[
v^{2} + 4v + 4
\]
**Step 3: (Optional) Factor the quadratic expression**
The quadratic \(v^{2} + 4v + 4\) can be factored further:
\[
v^{2} + 4v + 4 = (v + 2)^2
\]
**Final Answer:**
\[
v^{2} + 4v + 4
\]
Alternatively, factored form:
\[
(v + 2)^2
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
To simplify the expression \( \left(4 v^{4}+16 v^{3}+16 v^{2}\right) \div 4 v^{2} \), start by factoring out the common term from the numerator. This gives you: \[ = \frac{4 v^{2}(v^{2}+4 v+4)}{4 v^{2}}. \] The \( 4 v^{2} \) in the numerator and denominator cancels out, leading to: \[ = v^{2}+4 v+4. \] The resulting expression can be further factored as \( (v+2)^2 \). Now you've simplified the expression down to \( (v + 2)^2 \).
