Question
Given a polynomial and one of its factors, drag the remaining factors of the polynomial f(x)=x^3+3x^2-4;x-1
Answer
4.4
(301 Votes)
Kieran
Professional · Tutor for 6 years
Answer
Solution:1. Substitute x=1 into f(x) = x³ + 3x² - 4:Replace x = 1 into the given function: f(1) = (1**3) + 3*(1**2) - 4 = 1 + 3 - 4 = 0Thus f(1) = 0 verifying that x - 1 is indeed a factor of the given polynomial.2. Factor the given polynomial:We divide the original polynomial by the given factor(x-1) using synthetic division:``` 1 | 1 3 0 -4 | 1 4 4 |_________ 1 4 4 0```=> The quadratic polynomial f(x)/ (x - 1) = x² + 4x + 43. Find roots of the quadratic and thereby identifying remaining factors:We'll factorize f(x)/ (x - 1) . This can be written as: (x² + 4x + 4) = (x + 2)(x + 2)Every root of this quadratic represents a factor x - root :Hence, the remaining factors are x + 2, x + 2.In conclusion: `f(x)=x³+3x²-4` fully factors as `(x-1) * (x+2) * (x+2)`, illustratively broken down as: [x - 1], [x + 2], [x + 2].
Explanation
This math problem is in the domain of polynomial factoring.Given that `x-1` is a factor of `f(x)=x³+3x²-4`, and by the Factor Theorem if `x=a` is a root of a polynomial, then `x-a` is a factor of that polynomial.Consequently, if we substitute x=1 in the given expression `f(x) = x³ + 3x² - 4`, if `f(1)` equals 0, THEN ONLY `x - 1` is a factor of the given expression.So, first, we need to substitute x=1 into `f(x)` to confirm whether `x - 1` is truly a factor.If `x - 1` is indeed a factor, we have found one of the roots of the given polynomial equation, and divide the polynomial by this confirmed factor will give us the quadratic expression. Then we can find the final factors by solving the quadratic expression using the quadratic formula. Any roots from the quadratic can be made into factors by `x - rootVal`.