Part 1 of 2 Use synthetic division and the factor theorem to determine whether x-2 is a factor of f(x) f(x)=2x^3-9x^2+16x-12 Complete the first row of the synthetic division table square square

To determine whether \( x-2 \) is a factor of \( f(x) = 2x^3 - 9x^2 + 16x - 12 \) using synthetic division and the factor theorem, we need to follow these steps:<br /><br />1. **Set up the synthetic division table:**<br /> - Write down the coefficients of \( f(x) \): \( 2, -9, 16, -12 \).<br /> - Use the zero of the divisor \( x-2 \), which is \( 2 \).<br /><br />The synthetic division table will look like this:<br /><br />\[<br />\begin{array}{r|rrrr}<br />2 & 2 & -9 & 16 & -12 \\<br />\hline<br /> & & & & \\<br />\end{array}<br />\]<br /><br />2. **Perform the synthetic division:**<br /> - Bring down the first coefficient (2) directly below the line.<br /> - Multiply this number by the zero of the divisor (2) and write the result under the next coefficient.<br /> - Add the numbers in the second column and write the result below the line.<br /> - Continue this process for all columns.<br /><br />Here is the completed first row of the synthetic division table:<br /><br />\[<br />\begin{array}{r|rrrr}<br />2 & 2 & -9 & 16 & -12 \\<br />\hline<br /> & & 4 & & \\<br />\end{array}<br />\]<br /><br />So, the first row of the synthetic division table is:<br /><br />\[<br />\begin{array}{r|rrrr}<br />2 & 2 & -9 & 16 & -12 \\<br />\hline<br /> & & 4 & & \\<br />\end{array}<br />\]