factor f(x)=4x^(3)+19x^(2)-149x+36 into linear factors if f(x)=-9
after a few tries I found f(4) = 0
so (x-4) is a factor
by synthetic long division I got
(x-4)(4x^2 + 35x - 9)
= (x-4)(x+9)(4x - 1)
What does the fact that
f(x) = -9 have to do with it.?
Did you mean to say: f(9) = 0 , that would have given me the clue that x+9 is a factor.
To factorize the given polynomial f(x) = 4x^3 + 19x^2 - 149x + 36 into linear factors, we need to find the roots or zeros of the polynomial.
Step 1: Set f(x) equal to -9:
4x^3 + 19x^2 - 149x + 36 = -9
Step 2: Rewrite the equation in standard form by bringing all the terms to one side:
4x^3 + 19x^2 - 149x + 45 = 0
Step 3: To find the possible rational roots, we can use the rational root theorem. It states that if a polynomial has a rational root (p/q), then p must be a factor of the constant term (here, 45), and q must be a factor of the leading coefficient (here, 4).
The factors of 45 are ±1, ±3, ±5, ±9, ±15, and ±45. The factors of 4 are ±1 and ±2.
So, the possible rational roots are: ±1/1, ±3/1, ±5/1, ±9/1, ±15/1, ±45/1, ±1/2, ±3/2, ±5/2, ±9/2, ±15/2, ±45/2.
Step 4: Use synthetic division or substitute the possible rational roots back into the equation to see if any of them are roots of the polynomial. By checking each of the possible rational roots, we can find that x = -1 and x = -3/2 are the two roots of the given polynomial.
Step 5: Once we have found the roots, we can express the original polynomial as a product of linear factors.
The roots we found were x = -1 and x = -3/2. So, the linear factors are (x + 1) and (2x + 3).
Therefore, the factored form of f(x) = 4x^3 + 19x^2 - 149x + 36 when f(x) = -9 is:
f(x) = (x + 1)(2x + 3)