Question
Real Tutor Solution
Tutor-Verified Answer
Answer
The polynomial \( 3x^{4} - 3x^{2} - 27x^{2} + 27 \) factors to \( 3(x - 3)(x + 3)(x - 1)(x + 1) \).
Solution
To factorize the polynomial \( 3x^{4} - 3x^{2} - 27x^{2} + 27 \), follow these steps:
1. **Combine Like Terms:**
\[
3x^{4} - 3x^{2} - 27x^{2} + 27 = 3x^{4} - 30x^{2} + 27
\]
2. **Factor Out the Greatest Common Factor (GCF):**
\[
3x^{4} - 30x^{2} + 27 = 3(x^{4} - 10x^{2} + 9)
\]
3. **Factor the Quartic Polynomial:**
Let \( y = x^{2} \). Then the expression becomes:
\[
y^{2} - 10y + 9
\]
This quadratic factors as:
\[
y^{2} - 10y + 9 = (y - 1)(y - 9)
\]
Substitute back \( y = x^{2} \):
\[
(x^{2} - 1)(x^{2} - 9)
\]
4. **Factor Further Using the Difference of Squares:**
\[
x^{2} - 1 = (x - 1)(x + 1) \\
x^{2} - 9 = (x - 3)(x + 3)
\]
5. **Combine All Factors:**
\[
3(x - 1)(x + 1)(x - 3)(x + 3)
\]
**Final Factored Form:**
\[
3(x - 3)(x + 3)(x - 1)(x + 1)
\]
Answer: After factoring, the polynomial is 3 (x – 3)(x + 3)(x – 1)(x + 1)
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Mind Expander
To factorise the expression \( 3x^{4} - 3x^{2} - 27x^{2} + 27 \), let's first combine like terms. The expression simplifies to: \( 3x^{4} - 30x^{2} + 27 \). Next, we can factor out a common factor of 3: \( 3(x^{4} - 10x^{2} + 9) \). Now, let’s set \( y = x^{2} \), which transforms the quadratic into: \( 3(y^{2} - 10y + 9) \). Now we can factor the quadratic: \( y^{2} - 10y + 9 = (y - 1)(y - 9) \). Substituting back for \( y \): \( 3((x^{2} - 1)(x^{2} - 9)) \). Recognising the difference of squares, \( x^{2} - 1 = (x - 1)(x + 1) \) and \( x^{2} - 9 = (x - 3)(x + 3) \): So, the final factorised form is: \( 3(x - 1)(x + 1)(x - 3)(x + 3) \).
