Consider the polynomial p(x)=7x3+4x2−112x−64.
Part A: What is the complete factorization of p(x)=7x3+4x2−112x−64 over the integers?
Part B: What methods are used to factor p(x)=7x3+4x2−112x−64?
Select one answer for Part A and select all answers that apply for Part B.
A: (7x+4)(x2−16)
B: perfect-square trinomial
A: (7x+4)(x−4)(x+4)
B: grouping
B: difference of squares
A: (7x+4)(49x2+28x+16)
A: (7x+4)(x2+56x+16)
B: difference of cubes
My answer is
A: (7x+4)(x2+56x+16)
B: difference of squares
Actually, your answer for Part A is incorrect. Let me explain how to correctly find the complete factorization of the polynomial p(x)=7x^3+4x^2−112x−64 over the integers.
To factor the polynomial p(x), we can start by looking for any common factors among the coefficients of the terms. In this case, none of the coefficients have a common factor other than 1.
Next, we need to consider the possible rational roots of the polynomial. According to the Rational Root Theorem, any rational root of p(x) must be of the form p/q, where p is a factor of the constant term -64 and q is a factor of the leading coefficient 7.
The factors of -64 are ±1, ±2, ±4, ±8, ±16, ±32, ±64, and the factors of 7 are ±1, ±7. Therefore, the possible rational roots are:
±1/1, ±2/1, ±4/1, ±8/1, ±16/1, ±32/1, ±64/1, ±1/7, ±2/7, ±4/7, ±8/7, ±16/7, ±32/7, ±64/7
We can test each of these values by applying synthetic division to see if it produces a remainder of 0.
By testing these values, we find that -4 is a root of p(x). Therefore, (x+4) is a factor of p(x) and we can use synthetic division again to divide p(x) by (x+4), resulting in:
p(x) = (x+4)(7x^2-24x-16)
Now we have a quadratic expression 7x^2-24x-16 that can be factored further. In this case, the quadratic expression cannot be factored easily using common methods such as factoring or completing the square. We can instead use the quadratic formula to find the roots of the quadratic equation 7x^2-24x-16 = 0:
x = (-(-24) ± sqrt((-24)^2 -4(7)(-16))) / (2(7))
Simplifying this expression, we find that x ≈ -1.14 and x ≈ 2.43.
Therefore, the complete factorization of p(x) over the integers is:
p(x) = (x+4)(x+1.14)(x-2.43)
Now, let's move on to Part B of the question, which asks about the methods used to factor p(x)=7x^3+4x^2−112x−64.
The methods used to factor p(x) include:
1. Rational Root Theorem: Used to find the possible rational roots of p(x) by examining the factors of the constant term and leading coefficient.
2. Synthetic division: Used to test whether a possible rational root of p(x) is actually a root, and to divide p(x) by a factor once it is found.
3. Quadratic formula: Used to find the roots of the quadratic equation obtained from the factored quadratic expression, if needed.
Therefore, the correct answers for Part B are:
- Rational Root Theorem
- Synthetic division
- Quadratic formula
I hope this helps clarify the correct answers and the process for finding the factorization of the polynomial p(x).