Factor completely.
x, cubed, minus, 7, x, squared, minus, 13, x, plus, 91
x
3
−7x
2
−13x+91
The expression can be factored as: (x - 1)(x - 7)(x + 13)
To factor the expression x^3 - 7x^2 - 13x + 91 completely, you can use the Rational Root Theorem combined with synthetic division.
Step 1: List all possible rational roots.
The possible rational roots of this expression can be found by taking the factors of the constant term (91) divided by the factors of the leading coefficient (1). The factors of 91 are ±1, ±7, ±13, and ±91, and the factors of 1 are ±1. Therefore, the possible rational roots are ±1, ±7, ±13, and ±91.
Step 2: Test the possible rational roots using synthetic division.
We will test each possible root one by one using synthetic division to find the roots that make the polynomial equal to zero.
Testing x = 1:
1 | 1 -7 -13 91
| 1 -6 -19
____________
1 -6 -19 72
The remainder is 72, so x = 1 is not a root.
Testing x = -1:
-1 | 1 -7 -13 91
| -1 8 5
____________
1 -8 -5 96
The remainder is 96, so x = -1 is not a root.
Testing x = 7:
7 | 1 -7 -13 91
| 7 0 -91
___________
1 0 -13 0
The remainder is 0, so x = 7 is a root.
Step 3: Use the found root to write the factors.
(x - 7) is a factor of the expression.
Step 4: Divide the original polynomial by the factor.
Performing synthetic division with (x - 7) as a factor:
7 | 1 0 -13 0
| 7 49 252
______________
1 7 36 252
The quotient is x^2 + 7x + 36.
Step 5: Factor the quotient using either factoring, completing the square, or the quadratic formula.
The quadratic equation x^2 + 7x + 36 cannot be factored further, so it remains as it is.
Step 6: Write the completely factored form.
Putting everything together, the completely factored form of x^3 - 7x^2 - 13x + 91 is:
(x - 7)(x^2 + 7x + 36)
To factor the expression completely: x^3 - 7x^2 - 13x + 91, we can use various factoring techniques.
Step 1: Look for common factors among the terms. In this case, there are no common factors.
Step 2: Check if the expression can be factored using any special products or formulas. In this case, there is no apparent special product or formula to factor the given expression.
Step 3: Use synthetic division or guess and check method to find possible factors that make the expression equal to zero. For a cubic expression like this, the possible rational roots can be determined using the Rational Root Theorem. The possible rational roots in this case are the factors of the constant term (±1, ±7, ±13, ±91) divided by the factors of the leading coefficient (±1).
Let's try x = 1 as a possible root using synthetic division:
1 | 1 -7 -13 91
| 1 -6 -19
___________________________
1 -6 -19 72
Since the remainder is not zero, x = 1 is not a root.
Let's try x = -1 as a possible root using synthetic division:
-1 | 1 -7 -13 91
| -1 8 5
___________________________
1 -8 -5 96
Since the remainder is not zero, x = -1 is not a root.
Continue this process for all possible rational roots until a factor is found. In this case, after going through all possible rational roots, we find that none of them are factors of the given expression.
Therefore, the expression x^3 - 7x^2 - 13x + 91 cannot be factored further over the set of rational numbers. It is already in its simplest factored form.