if f(x)=x-1 and g(x)=4^4+x^3+x^2+1, find f(x) times g(x)

To find \(f(x) \times g(x)\) where \(f(x) = x - 1\) and \(g(x) = 4^4 + x^3 + x^2 + 1\), we need to multiply the two polynomials together.

Given:
\[ f(x) = x - 1 \]
\[ g(x) = 4^4 + x^3 + x^2 + 1 \]

First, simplify \(4^4\):
\[ 4^4 = 256 \]

So, we rewrite \(g(x)\):
\[ g(x) = 256 + x^3 + x^2 + 1 \]

Now we multiply \(f(x)\) by \(g(x)\):
\[ (x - 1)(256 + x^3 + x^2 + 1) \]

### Distribute each term of \(f(x)\) over \(g(x)\):

1. Multiply \(x\) by each term in \(g(x)\):
   \[
   x \cdot 256 = 256x
   \]
   \[
   x \cdot x^3 = x^4
   \]
   \[
   x \cdot x^2 = x^3
   \]
   \[
   x \cdot 1 = x
   \]

2. Multiply \(-1\) by each term in \(g(x)\):
   \[
   -1 \cdot 256 = -256
   \]
   \[
   -1 \cdot x^3 = -x^3
   \]
   \[
   -1 \cdot x^2 = -x^2
   \]
   \[
   -1 \cdot 1 = -1
   \]

### Combine all terms:

\[
256x + x^4 + x^3 + x - 256 - x^3 - x^2 - 1
\]

### Simplify by combining like terms:

\[
x^4 + (x^3 - x^3) + (-x^2) + (256x + x) - 256 - 1
\]

\[
x^4 - x^2 + 257x - 257
\]

So, the product \(f(x) \times g(x)\) is:

\[
\boxed{x^4 - x^2 + 257x - 257}
\]

Would you like details or have any questions about this solution?

Here are some related questions:

1. What is the degree of the polynomial \(x^4 - x^2 + 257x - 257\)?
2. How do you find the roots of a polynomial?
3. Can you differentiate the product \(f(x) \times g(x)\)?
4. How would you find the value of \(f(x) \times g(x)\) at \(x = 1\)?
5. What are the coefficients of each term in the polynomial \(x^4 - x^2 + 257x - 257\)?
6. How do you multiply polynomials using the distributive property?
7. What is the significance of the constant term in a polynomial?
8. Can the polynomial \(x^4 - x^2 + 257x - 257\) be factored further?

**Tip:** When simplifying polynomials, always combine like terms carefully to avoid errors.

Learn how to multiply polynomials f(x) = x - 1 and g(x) = 256 + x^3 + x^2 + 1. This problem involves distributing terms and combining like terms to find the product f(x) times g(x). Suitable for high school students and those studying algebraic expressions and polynomial multiplication.

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