Factor completely 4×4−24×3+36×2.
A) 4×2(x−3)2
B) 4(x−3)2
C) 4×2(x+3)(x−3)
D) 4x(x2−6x+9)
The Correct Answer and Explanation is:
To factor the expression ( 4x^4 – 24x^3 + 36x^2 ) completely, we start by identifying the common factors in each term.
Step 1: Identify Common Factors
The given polynomial consists of three terms:
- ( 4x^4 )
- ( -24x^3 )
- ( 36x^2 )
We observe that each term has a factor of ( 4x^2 ). Thus, we can factor ( 4x^2 ) out of the entire expression:
[
4x^2(x^2 – 6x + 9)
]
Step 2: Factor the Quadratic Expression
Next, we need to factor the quadratic expression ( x^2 – 6x + 9 ). To do this, we look for two numbers that multiply to ( 9 ) (the constant term) and add to ( -6 ) (the coefficient of ( x )). The numbers ( -3 ) and ( -3 ) satisfy both conditions because:
[
-3 \times -3 = 9
]
[
-3 + -3 = -6
]
Thus, we can rewrite the quadratic as:
[
x^2 – 6x + 9 = (x – 3)(x – 3) = (x – 3)^2
]
Step 3: Combine the Factors
Now substituting back into the expression, we have:
[
4x^2(x^2 – 6x + 9) = 4x^2(x – 3)^2
]
Conclusion
Thus, the completely factored form of the original expression ( 4x^4 – 24x^3 + 36x^2 ) is:
[
\boxed{4x^2(x – 3)^2}
]
Verification
To verify, we can expand ( 4x^2(x – 3)^2 ) back to see if we obtain the original expression:
- First, expand ( (x – 3)^2 ):
[
(x – 3)(x – 3) = x^2 – 6x + 9
]
- Now multiply by ( 4x^2 ):
[
4x^2(x^2 – 6x + 9) = 4x^4 – 24x^3 + 36x^2
]
Since the expansion matches the original expression, the factoring is confirmed to be correct. The answer is option A: ( 4x^2(x – 3)^2 ).