Question

Suppose \(f(x)=x^{3}+x^{2}-16 x-16\) and \(g(x)=x^{2}-4\) Find the zeros of \((f \circ g)(x)\)

Short answer

The zeros of \((f \, \circ \, g)(x)\) are the values of x that satisfy the polynomial \(x^6 - 11x^4 + 24x^2 - 32 = 0\).

Step-by-step solution

- Understand the composition of functions

The given functions are \(f(x) = x^3 + x^2 - 16x - 16\) and \(g(x) = x^2 - 4\). We are asked to find the zeros of \((f \, \circ \, g)(x)\), which means we need to find where \(f(g(x)) = 0\).

- Substitute \(g(x)\) into \(f(x)\)

Substitute \(g(x)\) into \(f(x)\) to get \(f(g(x))\). Thus, \(f(g(x)) = f(x^2 - 4)\). Now replace every \(x\) in \(f(x)\) with \(x^2 - 4\).

- Calculate \(f(x^2 - 4)\)

Replace \(x\) in \(f(x)\) with \(x^2 - 4\): \[ f(x^2 - 4) = (x^2 - 4)^3 + (x^2 - 4)^2 - 16(x^2 - 4) - 16 \]. Now expand and simplify the expression.

- Simplify the expression

Expand the terms one by one:1. \( (x^2 - 4)^3 = x^6 - 12x^4 + 48x^2 - 64 \)2. \( (x^2 - 4)^2 = x^4 - 8x^2 + 16 \)Substituting these into the expression, we get:\[ f(x^2 - 4) = (x^6 - 12x^4 + 48x^2 - 64) + (x^4 - 8x^2 + 16) - 16(x^2 - 4) - 16 \]This simplifies to:\[x^6 - 11x^4 + 24x^2 - 32 \]

- Set \(f(x^2 - 4) = 0\)

We need to find the zeros of \(x^6 - 11x^4 + 24x^2 - 32 = 0 \). Set each factor to zero and solve for x.

- Solve for \(x\)

Solve \(x^6 - 11x^4 + 24x^2 - 32 = 0 \) by letting \(u = x^2\): \(u^3 - 11u^2 + 24u - 32 = 0\). Solving for u, we get the values that satisfy this polynomial. Convert these solutions back to \(x\) by solving \(u = x^2\).

Key concepts

Finding Zeros

To find the zeros of a function means to find the values of the variable that make the function equal to zero. For the function \(f(x) = x^3 + x^2 - 16x - 16\), we need to find the x-values for which \(f(x) = 0\). When dealing with composed functions, like \(f(g(x))\), we substitute one function into another and then set the resulting expression equal to zero.

• Given \(f(x)=x^3+x^2-16x-16\) and \(g(x)=x^2-4\), our goal is to find where the composition \(f(g(x)) = 0\).
• Substituting \(g(x)\) into \(f(x)\), we get \(f(g(x)) = f(x^2 - 4)\).
• Next, we expand and simplify the expression to get it in a solvable form.

This might involve algebraic manipulation such as expanding polynomials, combining like terms, and simplifying. The ultimate goal is to solve the equation for \(x\).

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The general form of a polynomial in one variable, x, is###:

\[ P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \].

• Each term in the polynomial is called a 'monomial'.
• The highest power of the variable is called the 'degree' of the polynomial.
• Examples include \(x^2 - 4\) (a quadratic polynomial, degree 2) and \(x^3 + x^2 - 16x - 16\) (a cubic polynomial, degree 3).

When dealing with polynomial functions, operations like addition, subtraction, multiplication, and composition can be performed. The solution often requires manipulating these polynomials and using algebraic techniques to simplify them.

Algebraic Substitution

Algebraic substitution is a fundamental technique in algebra where one expression is replaced by another equivalent expression. This is particularly useful in simplifying and solving equations.

• For the given problem, we substitute \(g(x)\) into \(f(x)\) to create a new function: \(f(g(x))\).
• In the specific case here, we substitute \(g(x) = x^2 - 4\) into \(f(x) = x^3 + x^2 - 16x - 16\), giving us \( f(x^2 - 4) \).

This may involve:

• Expanding expressions such as \[ (x^2 - 4)^3 = x^6 - 12x^4 + 48x^2 - 64 \].
• Combining and simplifying like terms.
• Finally, solving the resulting polynomial equation, which might involve setting a new variable (e.g., \(u = x^2\)) to simplify the equation further.

Substitution allows us to handle more complex expressions by breaking them down into simpler parts.