factor fully:
x^3 + x^2 - 7x + 2 give n that x=2 is a root
To factor the given polynomial, we'll make use of the root given, which is x = 2.
Step 1: Verify if x = 2 is indeed a root of the polynomial.
To check this, we substitute x = 2 into the polynomial and see if it equals to zero.
Substituting x = 2 into the polynomial:
(2)^3 + (2)^2 - 7(2) + 2 = 8 + 4 - 14 + 2 = 0
Since the value is equal to zero, we confirm that x = 2 is a root of the polynomial.
Step 2: Synthetic division
To perform synthetic division, we divide the polynomial by x - 2 (since x = 2 is a root).
2 │ 1 1 -7 2
──────────
1 3 -1 0
The result of the synthetic division is 1x^2 + 3x - 1.
Step 3: Factor the remaining quadratic expression.
The synthetic division result, 1x^2 + 3x - 1, can be factored further. We'll use the factoring method or quadratic formula to find the roots.
Using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
Given a = 1, b = 3, and c = -1, we can substitute these values into the quadratic formula to find the roots.
x = (-3 ± √(3^2 - 4(1)(-1))) / 2(1)
x = (-3 ± √(9 + 4)) / 2
x = (-3 ± √13) / 2
Therefore, the roots of the quadratic expression 1x^2 + 3x - 1 are (-3 + √13) / 2 and (-3 - √13) / 2.
To summarize, the fully factored form of the polynomial x^3 + x^2 - 7x + 2 is:
(x - 2)(x - (-3 + √13) / 2)(x - (-3 - √13) / 2)