Factor and find all zeros. One factor has been given.
f(x)= x^3+9x^2+23x+15; x+5
divide the expression by the factor
that will leave a quadratic that you can use the formula on
To find the remaining factors of the given polynomial, we can use long division or synthetic division.
Using synthetic division:
First, we will write the given factor, x + 5, in the form (x - a), where 'a' is the opposite sign of 5. Therefore, 'a' is -5.
Now, set up the synthetic division as follows:
-5 | 1 9 23 15
Bring down the first coefficient:
1
Multiply the divisor (-5) by the first value brought down (1) and write the result under the next coefficient:
1 -5
Add the values in the second column:
1 -5 14
Multiply the divisor (-5) by the sum obtained (14), and write the result under the next coefficient:
1 -5 14 -70
Add the values in the second column:
1 -5 14 -55
The result (1 - 5x + 14x - 55) represents the polynomial after dividing by x + 5.
Hence, the factored form becomes:
f(x) = (x + 5)(x^2 - 5x + 14)
To find the remaining zeros, we need to solve the quadratic equation:
x^2 - 5x + 14 = 0
Using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Where a = 1, b = -5, and c = 14.
Now, substitute the values into the formula:
x = (-(-5) ± √((-5)^2 - 4(1)(14))) / (2*1)
Simplifying:
x = (5 ± √(25 - 56)) / 2
x = (5 ± √(-31)) / 2
Since we have a square root of a negative value, the solutions are complex numbers.
Therefore, the remaining zeros are:
x = (5 + √(-31)) / 2
x = (5 - √(-31)) / 2
So, the factors of the given polynomial are:
(x + 5)(x - [(5 + √(-31)) / 2])(x - [(5 - √(-31)) / 2])
To factor the given polynomial using the provided factor x+5, you can use synthetic division or long division. Since synthetic division is a quicker method, let's use that.
Step 1: Set up the synthetic division table:
Write down the coefficients of the polynomial in descending order.
-5 | 1 9 23 15
Step 2: Perform the synthetic division:
Bring down the coefficient of the highest degree term (1) to the bottom row of the table.
Multiply -5 by 1 to get -5, and write it below the coefficient of the second highest degree term (9).
Add the two numbers to get 4, and write this result below the coefficient of the third highest degree term (23).
Repeat the process until you reach the constant term (15).
-5 | 1 9 23 15
-5 -20 -15
___________________
1 4 3 0
Step 3: Interpret the results:
The result of the synthetic division gives us the quotient and the remainder. The quotient represents the coefficients of the lower-degree polynomial, and the remainder represents the constant term.
In this case, we have:
Quotient: 1 4 3
Remainder: 0
Step 4: Write the factored form of the polynomial:
The factored form of the polynomial with the given factor x+5 is obtained by inserting the quotient back into the equation.
f(x) = (x+5)(x^2+4x+3)
Now we can find the zeros of the function.
To find the zeros, we set the factored form of the polynomial equal to zero and solve for x:
(x+5)(x^2+4x+3) = 0
Setting each factor to zero:
x+5 = 0 -> x = -5
x^2+4x+3 = 0
Using factoring or the quadratic formula, we can solve the quadratic equation:
x^2+4x+3 = 0
(x+1)(x+3) = 0
Setting each factor to zero:
x+1 = 0 -> x = -1
x+3 = 0 -> x = -3
So, the zeros of the function f(x) = x^3+9x^2+23x+15 are: x = -5, x = -1, x = -3.