Given f(x)=3x^3+8x^2+27x+72 is a polynomial with f(−8/3)=0 , factor f(x) completely.(1 point)
1. f(x)=(3x+8)(x+3)(x-3)
2. f(x)=(x-8/3)(3x^2+16x+209/3)
3. f(x)=3(x-8/3)(x^2+9)
4. f(x)=(3x+8)(x^2+9)
The correct answer is:
3. f(x)=3(x-8/3)(x^2+9)
To factor the polynomial f(x) completely, we need to find the factors that satisfy f(-8/3) = 0.
Substituting x = -8/3 into f(x), we have:
f(-8/3) = 3(-8/3)^3 + 8(-8/3)^2 + 27(-8/3) + 72
Simplifying this expression, we find:
f(-8/3) = -512/27 + 256/9 - 216/3 + 72
f(-8/3) = -512/27 + 768/27 - 648/27 + 72
f(-8/3) = -512/27 + 768/27 - 648/27 + 1944/27
f(-8/3) = 1552/27
Since f(-8/3) = 0, the factor (x + 8/3) is present in f(x).
To factor f(x) completely, we divide the polynomial by (x + 8/3). The resulting quotient will give us a quadratic expression that we can further factor.
Using polynomial long division:
3x^2 + 16x + 209/3
_________________________
x + 8/3 | 3x^3 + 8x^2 + 27x + 72
We find that the quotient is 3x^2 + 16x + 209/3.
Now we can factor 3x^2 + 16x + 209/3 by factoring the quadratic expression. However, since this expression does not factor easily (using the factoring methods), we can leave it as is:
f(x) = (x + 8/3)(3x^2 + 16x + 209/3)
Therefore, the correct answer is:
2. f(x) = (x - 8/3)(3x^2 + 16x + 209/3)
To factor the polynomial f(x)=3x^3+8x^2+27x+72 completely, we can use the given zero, f(-8/3)=0, to find one of the linear factors.
Since f(-8/3)=0, it means that x=-8/3 is a root of the polynomial. To factor it out, we can use the factor theorem. According to the factor theorem, if x=c is a root of the polynomial, then (x-c) is a factor of the polynomial.
So, we can factor out (x+8/3) from f(x) by dividing f(x) by (x+8/3) using polynomial long division or synthetic division. Here's the polynomial division:
_____
(x+8/3) | 3x^3 + 8x^2 + 27x + 72
- (3x^3 + 8x^2)
____________
19x + 72
- (19x + 72)
____________
0
We see that (x+8/3) divides evenly into f(x), leaving us with 19x + 72.
Therefore, f(x) can be factored as:
f(x) = (x+8/3)(19x + 72)
Now, we have factored out one linear factor. To further factorize f(x), let's see if there are any quadratic factors.
We can use the quadratic formula to determine if f(x) has any real roots for the remaining quadratic factor (19x + 72).
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / (2a),
where in our case:
a = 19
b = 0
c = 72
Let's calculate the discriminant (b^2 - 4ac) to determine if there are any real roots.
Discriminant = (0^2) - (4 * (19) * (72)) = -5472
Since the discriminant is negative (-5472 < 0), it means that the quadratic factor 19x + 72 does not have any real roots. Therefore, it cannot be further factored.
Hence, the complete factorization of f(x) is:
f(x) = (x+8/3)(19x + 72)
Therefore, the correct answer is:
2. f(x) = (x-8/3)(3x^2 + 16x + 209/3)