Use synthetic division and the factor theorem to determine wheth f(x)=4x^3-12x^2+13x-10 Complete the first row of the synthetic division table. square square square

To use synthetic division and the factor theorem to determine whether \( f(x) = 4x^3 - 12x^2 + 13x - 10 \) has any rational roots, we first need to identify possible rational roots using the Rational Root Theorem. The Rational Root Theorem states that any rational root of the polynomial \( f(x) \) is a factor of the constant term divided by a factor of the leading coefficient.<br /><br />For \( f(x) = 4x^3 - 12x^2 + 13x - 10 \):<br />- The constant term is \(-10\).<br />- The leading coefficient is \(4\).<br /><br />The factors of \(-10\) are \(\pm 1, \pm 2, \pm 5, \pm 10\).<br />The factors of \(4\) are \(\pm 1, \pm 2, \pm 4\).<br /><br />Thus, the possible rational roots are:<br />\[ \pm 1, \pm \frac{1}{2}, \pm 2, \pm 5, \pm 10 \]<br /><br />We will test these possible roots using synthetic division to see if they yield a remainder of zero, which would indicate that they are indeed roots of the polynomial.<br /><br />Let's start with \( x = 1 \):<br /><br />1. Write down the coefficients of \( f(x) \): \( 4, -12, 13, -10 \).<br />2. Set up the synthetic division table with \( x = 1 \):<br /><br />\[<br />\begin{array}{r|rrrr}<br />1 & 4 & -12 & 13 & -10 \\<br />\hline<br /> & & & & \\<br />\end{array}<br />\]<br /><br />Now, perform the synthetic division steps:<br /><br />1. Bring down the first coefficient (4) directly below the line.<br />2. Multiply this number by \( x = 1 \) and write the result under the next coefficient.<br />3. Add the numbers in the second column and write the result below the line.<br />4. Repeat the process for each column.<br /><br />\[<br />\begin{array}{r|rrrr}<br />1 & 4 & -12 & 13 & -10 \\<br />\hline<br /> & & 4 & -8 & 5 & -5 \\<br />\end{array}<br />\]<br /><br />The final row represents the coefficients of the quotient polynomial and the remainder. Since the remainder is \(-5\), \( x = 1 \) is not a root of the polynomial.<br /><br />We can continue this process with other possible rational roots to find any actual roots. However, for the purpose of completing the first row of the synthetic division table, we have shown the setup for testing \( x = 1 \).