Respuesta :
x = 1/3 x= -3
Step-by-step explanation:
For the function f(x) = 3x^3 + 11x^2 + 5x - 3:
* Factors of the constant term (-3): 1, 3
* Factors of the leading coefficient (3): 1, 3
* Possible rational roots: Based on the factors, the possible rational roots are ±1, ±3
Steps to factor and find all roots:
* List the possible rational roots: As identified earlier, the possible rational roots are ±1 and ±3.
* Try synthetic division: Start with a possible rational root (let's try x = 1). If the remainder is 0, then x = 1 is a root.
* Repeat synthetic division: If x = 1 is not a root, repeat the process with other possible rational roots until a root is found.
* Factor the expression: Once a root is found, use it to factor the expression. This will potentially lead to finding other roots.
* Repeat for remaining factors: If the factored expression is not completely factored, repeat the process of finding roots and factoring until the expression is completely factored.
Example of synthetic division (for x = 1):
3 11 5 -3
1 | 3 12 7 4
-- | -------------
3 13 12 1
Since the remainder is not 0, x = 1 is not a root. You can repeat this process for other possible rational roots to find the actual roots and factors of the polynomial.
Okay, let's go through this step-by-step:
2. f(x) = 3x^3 + 11x^2 + 5x - 3
Factors of the constant (-3): -1, 3
Factors of the leading coefficient (3): -3, 1
Possible rational roots: -1, 3, -1/3, 1/3
To factor:
(3x - 1)(x^2 + 3x + 3)
Set each factor equal to 0 to find roots:
3x - 1 = 0
x = 1/3
x^2 + 3x + 3 = 0
Discriminant b^2 - 4ac = 0, no real roots
Therefore the roots are:
x = 1/3
To apply the Rational Roots Theorem, we first need to list all possible factors of the constant term and the leading coefficient. For \( f(x) = 3x^3 + 11x^2 + 5x - 3 \):
1. Factors of the constant term (-3): ±1, ±3
2. Factors of the leading coefficient (3): ±1, ±3
Possible rational roots are all the possible combinations of factors of the constant term divided by factors of the leading coefficient:
±1, ±1/3, ±3, ±3/3, ±1, ±3, ±3, ±1
These simplify to:
±1, ±1/3, ±3, ±1, ±3, ±1, ±3
Now, let's use synthetic division to check each possible root:
1. \( x = 1 \):
- \( 3(1)^3 + 11(1)^2 + 5(1) - 3 = 3 + 11 + 5 - 3 = 16 \) (not 0)
2. \( x = -1 \):
- \( 3(-1)^3 + 11(-1)^2 + 5(-1) - 3 = -3 + 11 - 5 - 3 = 0 \)
- Remainder is 0, so \( x + 1 \) is a factor.
3. Now, perform synthetic division with \( x + 1 \) to find the other factor:
- \( 3x^2 + 8x - 3 \)
- Possible roots are ±1, ±1/3, ±3, ±1, ±3, ±1, ±3
- Let's try \( x = 1 \):
- \( 3(1)^2 + 8(1) - 3 = 3 + 8 - 3 = 8 \) (not 0)
- Let's try \( x = -1 \):
- \( 3(-1)^2 + 8(-1) - 3 = 3 - 8 - 3 = -8 \) (not 0)
- Let's try \( x = 3 \):
- \( 3(3)^2 + 8(3) - 3 = 27 + 24 - 3 = 48 \) (not 0)
- Let's try \( x = -3 \):
- \( 3(-3)^2 + 8(-3) - 3 = 27 - 24 - 3 = 0 \)
- Remainder is 0, so \( x + 3 \) is a factor.
4. Now, perform synthetic division with \( x + 3 \):
- \( 3x^2 + 8x - 3 \) divided by \( x + 3 \) gives \( 3x - 1 \)
- The factors are \( (x + 1)(3x - 1) \)
5. To find the roots, set each factor to zero:
- \( x + 1 = 0 \) gives \( x = -1 \)
- \( 3x - 1 = 0 \) gives \( x = \frac{1}{3} \)
So, the roots of \( f(x) = 3x^3 + 11x^2 + 5x - 3 \) are \( x = -1, \frac{1}{3} \), and \( x = 3 \).
