Problem 14 - Precalculus

Question

Determine whether the given value is a zero of the function. \(g(x)=x^{4}+8 x^{3}+9 x^{2}-8 x-10\) (a) \(x=1\) (b) \(x=\sqrt{6}-4\) (c) \(x=\sqrt{6}+4\)

Step-by-Step Solution

Answer

(a) Yes, (b) Yes, (c) No.

Step 1: Substitute x = 1 into the function

Substitute the given value \(x=1\) into the function \(g(x)=x^{4}+8 x^{3}+9 x^{2}-8 x-10\). Calculate each term separately:\[g(1) = 1^{4} + 8(1)^{3} + 9(1)^{2} - 8(1) - 10\] which simplifies to \[1 + 8 + 9 - 8 - 10 = 0\].Since \(g(1) = 0\), \(x=1\) is a zero of the function.

Step 2: Substitute x = \(\sqrt{6}-4\) into the function

Substitute \(x=\sqrt{6}-4\) into the function. First, calculate every component:\[(\sqrt{6} - 4)^{4}, 8(\sqrt{6} - 4)^{3}, 9(\sqrt{6} - 4)^{2}, -8(\sqrt{6} - 4), -10\].Substitute and simplify each component:\[(\sqrt{6} - 4)^{4} = 64 - 32\sqrt{6} + 24 - 4\sqrt{6}\]\[8((\sqrt{6} - 4)^3) = -192 + 96\sqrt{6} - 48 + 8\sqrt{6}\]Combine these terms to find that terms cancel each other, sum up to zero: \[g(\sqrt{6} - 4) = 0\]Thus, \(x=\sqrt{6}-4\) is a zero of the function.

Step 3: Substitute x = \(\sqrt{6}+4\) into the function

Substitute \(x=\sqrt{6}+4\) into the function. Calculate each necessary term:\[(\sqrt{6} + 4)^{4}, 8(\sqrt{6} + 4)^{3}, 9(\sqrt{6} + 4)^{2}, -8(\sqrt{6} + 4), -10\].When computing these terms, you will notice that they will not cancel each other out, and the sum is:\[g(\sqrt{6} + 4) eq 0\].Thus, \(x=\sqrt{6}+4\) is not a zero of the function.

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions consisting of variables and coefficients. These functions can have multiple terms, each made up of a variable raised to a power, usually known as the degree of the polynomial. The degree indicates the highest power of the variable.

• A polynomial with one variable can be expressed generally as: \[ f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \] where each \(a_i\) represents a coefficient.
• The highest exponent \(n\) tells us the degree of the polynomial.
• For instance, in \(g(x) = x^4 + 8x^3 + 9x^2 - 8x - 10\), the degree is 4.

Polynomial functions are continuous and smooth, making them crucial in mathematical modeling and real-world applications.
They are often used to represent the motion of objects, areas under curves, and even population growth predictions.

Evaluating Functions

Evaluating a function involves substituting a given number into the function and simplifying the result to find the output. This process helps us determine specific values of functions, including discovering zeros.

• To evaluate a polynomial function, substitute the value into every instance of the variable in the function.
• For example, if you evaluate \(g(x) = x^4 + 8x^3 + 9x^2 - 8x - 10\) at \(x = 1\), substitute 1 for \(x\):\[ g(1) = 1^4 + 8(1)^3 + 9(1)^2 - 8(1) - 10 \]Simplifying, this becomes \(0\).
• By determining the output, you can conclude whether a given value is a zero of the function.

Through systematic evaluation, potential zeros of a function can be confirmed, as they result in an output of zero when substituted.

Rational Roots

Rational roots are solutions to polynomial equations that can be expressed as a ratio of two integers. They are integral in determining zeros of polynomial functions, as zeros are the values where the function's output will be zero.

• Potential rational roots can often be guessed or tested using the Rational Root Theorem. This theorem suggests that any rational root, \(\frac{p}{q}\), of the polynomial \(a_n x^n + a_{n-1} x^{n-1} + ... + a_0\), where \(a_n\) and \(a_0\) are the coefficients of the highest-degree and constant terms respectively, can have a numerator \(p\) that is a factor of \(a_0\) and a denominator \(q\) that is a factor of \(a_n\).
• This theorem enables a check of basic rational values to predict a zero quickly, saving time and effort during evaluation.
• For example, checking if \(x = 1\) is a zero for our function \(g(x) = x^4 + 8x^3 + 9x^2 - 8x - 10\) is a use of inspecting rational values.

Though finding rational roots might not always provide all zeros of a polynomial, it assists in forming a foundation to identify at least some of them easily.