Which polynomial is factored completely
a. 4 (4x^4-1)
b. 2x (y^3-4y^2+5y)
c. 3x (9×^2+1)
d. 5x^2-17x+14?
The Correct Answer and Explanation is :
The correct answer is: d. ( 5x^2 – 17x + 14 )
To determine which polynomial is factored completely, let’s analyze each option one by one:
a. ( 4(4x^4 – 1) )
This expression can be factored further. The term ( 4x^4 – 1 ) is a difference of squares and can be factored as:
[
4x^4 – 1 = (2x^2 – 1)(2x^2 + 1)
]
Thus, the complete factorization is:
[
4(2x^2 – 1)(2x^2 + 1)
]
This means option (a) is not fully factored.
b. ( 2x(y^3 – 4y^2 + 5y) )
First, factor out the greatest common factor (GCF) in the polynomial:
[
y^3 – 4y^2 + 5y = y(y^2 – 4y + 5)
]
Now, we need to factor ( y^2 – 4y + 5 ). The discriminant is ( (-4)^2 – 4(1)(5) = 16 – 20 = -4 ), which is negative, indicating no real roots. Therefore, ( y^2 – 4y + 5 ) cannot be factored further with real numbers. Thus, the factorization is:
[
2xy(y^2 – 4y + 5)
]
This indicates option (b) is not completely factored.
c. ( 3x(9x^2 + 1) )
The term ( 9x^2 + 1 ) does not factor further over the reals because it is a sum of squares. Therefore, the factorization remains:
[
3x(9x^2 + 1)
]
This means option (c) is not completely factored.
d. ( 5x^2 – 17x + 14 )
To factor this quadratic polynomial, we need to find two numbers that multiply to ( 5 \times 14 = 70 ) and add to (-17). The numbers (-5) and (-14) satisfy these conditions. Therefore, we can factor it as:
[
5x^2 – 5x – 14x + 14 = 5x(x – 1) – 14(x – 1) = (5x – 14)(x – 1)
]
Thus, option (d) is fully factored.
Conclusion
The only polynomial that is factored completely is d. ( 5x^2 – 17x + 14 ), which factors to ( (5x – 14)(x – 1) ). Options a, b, and c can all be factored further, indicating that they are not in their completely factored forms.