Use synthetic division to find the zeroes of the function f(x) = x^3 + x^2 +4x+4
Need help on this we have a test when i go back to school please help this was an example given and i dont understand it.
To find the zeroes of the function f(x) = x^3 + x^2 + 4x + 4 using synthetic division, follow these steps:
Step 1: Write down the coefficients of the polynomial in the form of (constant term, coefficient of x^3, coefficient of x^2, coefficient of x).
The coefficients for f(x) = x^3 + x^2 + 4x + 4 are (4, 1, 1, 1).
Step 2: Write down the potential zeroes of the function. For this polynomial, the potential zeroes are the factors of the constant term divided by the factors of the coefficient of x^3.
The factors of the constant term 4 are ±1, ±2, ±4.
The factors of the coefficient of x^3 1 are ±1.
So, the potential zeroes are ±1, ±2, ±4.
Step 3: Choose a potential zero and apply synthetic division.
Let's try x = -1 as a potential zero:
-1 | 1 1 4 4
|______-1 0 -4
1 0 4 0
The numbers in the bottom row are the coefficients of the resulting quadratic equation after division. The final number represents the remainder.
The resulting quadratic equation in this case is x^2 + 4. So, the remainder is 0.
Step 4: Repeat steps 3 for the resulting quadratic equation until a zero is found.
In this case, the quadratic equation is x^2 + 4. Since this is a non-factorable quadratic equation with no real roots, we cannot find any potential zeroes.
Therefore, the zeroes of the function f(x) = x^3 + x^2 + 4x + 4 are not rational or real numbers.
Sure, I can help you with that! Synthetic division is a method used to find the zeroes of a polynomial function.
To begin, let's write down the coefficients of the polynomial:
f(x) = x^3 + x^2 + 4x + 4
Using synthetic division, you'll need to pick a value for x (which we call the "test" value) that makes the polynomial equal to zero. This test value will help us find one zero, and we can then factorize the polynomial further if necessary.
Let's start with a common test value of x = -1 and perform the synthetic division step by step:
Step 1: Write down the coefficients of the polynomial: 1, 1, 4, 4.
Step 2: Place the test value (-1) outside a division symbol and the first coefficient (1) inside the symbol.
-1 | 1 1 4 4
Step 3: Bring down the first coefficient (1).
-1 | 1
------
1
Step 4: Multiply the test value (-1) by the number at the bottom of the column and write it below the next coefficient in the row.
-1 | 1 1
------
1
-1
Step 5: Add the numbers in the second column and write the result below the line.
-1 | 1 1
------
1
-1
-----
0
Step 6: The result at the bottom is zero, which means that the test value -1 is a zero of the polynomial.
Now, we have factored the polynomial and obtained a quadratic equation:
f(x) = (x + 1) * (x^2 + 0x + 1)
The quadratic equation x^2 + 0x + 1 does not factor further. In this case, we can use the quadratic formula to find the remaining zeroes:
x = (-b ± √(b^2 - 4ac)) / (2a)
For x^2 + 0x + 1, a = 1, b = 0, and c = 1. Plugging these values into the quadratic formula, we get:
x = (-0 ± √(0^2 - 4(1)(1))) / (2(1))
x = ± √(-4) / 2
Since the expression √(-4) gives us imaginary numbers, there are no real solutions for the remaining zeroes. Therefore, the zeroes of the function f(x) = x^3 + x^2 + 4x + 4 are x = -1.
Remember to double-check your steps and calculations when using synthetic division. Good luck on your test!
a nice interactive calculator can be found below. Play around with it to see how things work.
http://www.mathportal.org/calculators/polynomials-solvers/synthetic-division-calculator.php