Questions: Use substitution to determine whether 3 is a zero of the function. f(x)=x^4-6x^3+4x^2-6x-63

Use substitution to determine whether 3 is a zero of the function. f(x)=x^4-6x^3+4x^2-6x-63
Transcript text: Use substitution to determine whether 3 is a zero of the function. \[ f(x)=x^{4}-6 x^{3}+4 x^{2}-6 x-63 \]
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Solution

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Solution Steps

To determine whether 3 is a zero of the function \( f(x) = x^4 - 6x^3 + 4x^2 - 6x - 63 \), we substitute \( x = 3 \) into the function and evaluate it. If the result is zero, then 3 is a zero of the function.

Step 1: Substitute \( x = 3 \) into the function

To determine if 3 is a zero of the function \( f(x) = x^4 - 6x^3 + 4x^2 - 6x - 63 \), we substitute \( x = 3 \) into the function:

\[ f(3) = 3^4 - 6 \times 3^3 + 4 \times 3^2 - 6 \times 3 - 63 \]

Step 2: Calculate each term

Calculate each term separately:

  • \( 3^4 = 81 \)
  • \( 6 \times 3^3 = 6 \times 27 = 162 \)
  • \( 4 \times 3^2 = 4 \times 9 = 36 \)
  • \( 6 \times 3 = 18 \)
Step 3: Evaluate the function

Substitute the calculated values back into the function:

\[ f(3) = 81 - 162 + 36 - 18 - 63 \]

Step 4: Simplify the expression

Simplify the expression by performing the arithmetic operations:

\[ f(3) = 81 - 162 + 36 - 18 - 63 = -126 \]

Final Answer

\(\boxed{-126}\)

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