Problem 90 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find all real zeros of the polynomial function. $$g(x)=4 x^{4}-11 x^{3}-22 x^{2}+8 x$$

Step-by-Step Solution

Answer

The real zeros of the function are x = 0, x = -1⁄2, and x = 4.

Step 1: Factor out common factor

Factor out the common factor which is x in this case, from the equation. So the equation is converted as, \(x(4x^{3} - 11x^{2} - 22x + 8) = 0\).

Step 2: Apply The Rational Root Theorem

The Rational Root Theorem gives us a list of possible rational roots of a polynomial function. This it helps us to get two zeros of this function as x = 0,-2. After doing synthetic division with each, we find that -2 is a root, allowing us to further reduce the polynomial to, \(x(2x^2 - 3x - 4) = 0\).

Step 3: Factor the polynomial

Factor the quadratic polynomial further to obtain \(x(2x +1)(x - 4) = 0\).

Step 4: Set each factor equal to zero

Now, each factor can be set to zero, giving the solutions to the equation as x = 0, x = 4 and x = -1⁄2.

Key Concepts

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial into simpler 'factor' polynomials that, when multiplied together, give you the original polynomial. When faced with a polynomial like
\( g(x) = 4x^4 - 11x^3 - 22x^2 + 8x \),
the first step is often to look for common factors among the terms. In the example, each term contains an
\( x \),
which can be factored out to simplify the polynomial to
\( x(4x^3 - 11x^2 - 22x + 8) \).
This technique can significantly decrease the complexity of the polynomial, making the subsequent factoring steps easier to handle. Factoring is a crucial step in finding the zeros of a polynomial because it sets the stage for using the Zero Product Property.

Rational Root Theorem

The Rational Root Theorem is a handy tool in finding the potential zeros of a polynomial with integer coefficients. For a polynomial
\( p(x) = a_nx^n + a_{n-1}x^{n-1} + \.\.\. + a_1x + a_0 \),
this theorem suggests that any rational zero, in the form of a fraction
\(\frac{p}{q}\),
must have
\(p\),
a factor of the constant term
\(a_0\),
and
\(q\),
a factor of the leading coefficient
\(a_n\).
In the given example, we used the Rational Root Theorem to identify potential rational zeros and then verifited through synthetic division which of these roots are actual zeros of polynomial
\( g(x) \). This process discovered that
\( -2 \)
is indeed a rational zero of the polynomial.

Synthetic Division

Synthetic division is a shortcut method, used to divide a polynomial by a binomial of the form
\( x - c \).
It involves fewer steps than the long division method, especially when dealing with polynomials. When we have a suspected zero from the Rational Root Theorem, synthetic division helps to confirm if that value is a true zero of the polynomial. For the polynomial
\( g(x) \),
synthetic division was used to check
\( -2 \),
and it turned out to be a zero because the remainder was zero, allowing us to further factor the polynomial. This process is pivotal for simplifying the polynomial to make it easier to find its real zeros.

Zero Product Property

The Zero Product Property states that if the product of two or more factors is zero, then at least one of the factors must be zero. After fully factoring the polynomial, we apply this property to the factors individually. In our example, after factoring, we end up with the factors
\( x \),

\( 2x+1 \),
and
\( x-4 \).
Setting each factor equal to zero gives the solutions to the equation:
\( x = 0 \),
\( x = -\frac{1}{2} \),
and
\( x = 4 \).
The Zero Product Property is crucial as it finally provides the real zeros of the polynomial after other methods have been used to factor it down to its simplest form.