In example 1, the polynomial x^3-13x^2+43x-15 was reduced to the factors x-5 and x^2-8x+3. Find the zeros of the polynomial.
The zeros of a polynomial are the values of x that make the polynomial equal to zero.
In this case, the two factors we obtained from the reduction are (x-5) and (x^2-8x+3).
To find the zeros, we set each factor equal to zero and solve for x.
For the factor x-5:
x - 5 = 0
x = 5
For the factor x^2-8x+3, we can use the quadratic formula to solve for x.
The quadratic formula is x = (-b ± √(b^2 - 4ac)) / (2a), given ax^2 + bx + c = 0.
In this case, a = 1, b = -8, and c = 3.
Substituting these values into the quadratic formula, we get:
x = (-(-8) ± √((-8)^2 - 4(1)(3))) / (2(1))
x = (8 ± √(64 - 12)) / 2
x = (8 ± √52) / 2
x = (8 ± 2√13) / 2
x = 4 ± √13
So, the zeros of the polynomial x^3-13x^2+43x-15 are x = 5, x = 4 + √13, and x = 4 - √13.
To find the zeros of a polynomial, we set it equal to zero and solve for x. Thus, for the given polynomial x^3 - 13x^2 + 43x - 15, we have:
x^3 - 13x^2 + 43x - 15 = 0
Next, we need to use the given factors to factorize the polynomial. From the given information, we have:
x^3 - 13x^2 + 43x - 15 = (x - 5)(x^2 - 8x + 3) = 0
Now, we can set each factor equal to zero and solve for x individually:
Setting (x - 5) = 0, we have x = 5 as one possible zero.
Setting (x^2 - 8x + 3) = 0, we need to solve this quadratic equation. We can either use factoring or the quadratic formula to find the remaining zeros. Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac))/(2a)
Here, a = 1, b = -8, and c = 3. Substituting these values into the quadratic formula, we get:
x = (-(-8) ± √((-8)^2 - 4(1)(3)))/(2(1))
x = (8 ± √(64 - 12))/2
x = (8 ± √52)/2
x = (8 ± 2√13)/2
x = 4 ± √13
Thus, the zeros of the polynomial x^3 - 13x^2 + 43x - 15 are x = 5, x = 4 + √13, and x = 4 - √13.
To find the zeros of a polynomial, we need to solve the equation when the polynomial is equal to zero. In this case, we are given that the polynomial has been reduced to the factors (x - 5) and (x^2 - 8x + 3).
So, we can set each factor equal to zero and solve for x separately.
1. Setting x - 5 = 0:
x = 5
2. Setting x^2 - 8x + 3 = 0:
To solve this quadratic equation, we can use factoring, completing the square, or the quadratic formula.
Factoring:
We need to find two numbers whose product is 3 and whose sum is -8. The numbers are -1 and -3.
So, we can rewrite the equation as:
(x - 1)(x - 3) = 0
Setting each factor equal to zero:
x - 1 = 0 --> x = 1
x - 3 = 0 --> x = 3
Therefore, the zeros of the polynomial x^3 - 13x^2 + 43x - 15 are x = 5, x = 1, and x = 3.