Question

divide ( x^{3}-3x^{2}-10x + 24 ) by ( x - 2 ). step 1: fill in the missing number. ( 2 \begin{array}{r} 1 - 3 - 10 24 \\ hline end{array} )

Explanation:

Step1: Use synthetic division

Set up synthetic division with \(x = 2\). The coefficients of the polynomial \(x^{3}-3x^{2}-10x + 24\) are \(1,-3,-10,24\).
Bring down the first coefficient \(1\).

Step2: Multiply and add

Multiply \(1\) by \(2\) to get \(2\). Add to the next coefficient: \(-3+2=-1\).
Multiply \(-1\) by \(2\) to get \(-2\). Add to the next coefficient: \(-10+( - 2)=-12\).
Multiply \(-12\) by \(2\) to get \(-24\). Add to the last coefficient: \(24+( - 24)=0\).

Answer:

The remainder is \(0\) (since the last sum in synthetic division is \(0\)), which means \(x - 2\) is a factor of \(x^{3}-3x^{2}-10x + 24\).