Use the rational zero theorem to find all rational zeros, and use those zeros to write
f(x) = 2x^4-x^3-9x^2+4x+4
To use the Rational Zero Theorem to find all the rational zeros of a polynomial, we need to consider the factors of the constant term divided by the factors of the leading coefficient.
The constant term of the given polynomial is 4, and the leading coefficient is 2. So, the potential rational zeros are the factors of 4 divided by the factors of 2. The factors of 4 are ±1, ±2, and ±4, while the factors of 2 are ±1 and ±2.
Therefore, the potential rational zeros are: ±1, ±2, ±4.
To determine if any of these potential zeros are actual zeros, we can use polynomial long division or synthetic division until we find a value that gives a remainder of zero.
Let's try dividing the polynomial by some of the potential rational zeros:
1. Testing f(1):
When we substitute x = 1 into the polynomial, we get:
f(1) = 2(1)^4 - (1)^3 - 9(1)^2 + 4(1) + 4
= 2 - 1 - 9 + 4 + 4
= 0
Since f(1) is zero, 1 is a zero of the polynomial.
2. Testing f(-1):
When we substitute x = -1 into the polynomial, we get:
f(-1) = 2(-1)^4 - (-1)^3 - 9(-1)^2 + 4(-1) + 4
= 2 + 1 - 9 - 4 + 4
= -6
This means f(-1) is not zero.
By continuing this process, we can verify that 1 is a zero of the polynomial, while none of the other potential zeros ±2 and ±4 are zeros.
Therefore, the rational zeros of f(x) = 2x^4 - x^3 - 9x^2 + 4x + 4 are x = 1.
We can write f(x) in factored form using the found zero:
f(x) = (x - 1)(2x^3 + x^2 - 8x - 4)
Note: The remaining factor, 2x^3 + x^2 - 8x - 4, can be further factored, but it contains no rational zeros. To find additional zeros, you can employ other methods like the Rational Root Theorem or quadratic factoring.