Question 3.3.1: Deciding Whether x − k Is a Factor Determine whether x - 1 i...
Deciding Whether x − k Is a Factor
Determine whether x – 1 is a factor of each polynomial.
(a) ƒ(x) = 2x^{4} + 3x² – 5x + 7
(b) ƒ(x) = 3x^{5} – 2x^{4} + x³ – 8x² + 5x + 1
(a) By the factor theorem, x – 1 will be a factor if ƒ(1) = 0. Use synthetic division and the remainder theorem to decide.
\begin{matrix} 1) \overline{\begin{matrix} 2 &0 &3&-5&7\end{matrix} } \\ \underline{\begin{matrix} &&2 & 2& &5&0\end{matrix} } \\ \begin{matrix} 2 & 2 & 5& 0& 7\end{matrix}&← ƒ(1) = 7\end{matrix}
The remainder is 7, not 0, so x – 1 is not a factor of 2x^{4} + 3x² – 5x + 7.
(b) \begin{matrix} 1) \overline{\begin{matrix} 3&-2 &1&-8&5& 1\end{matrix} }&←ƒ(x) = 3x^{5} – 2x^{4} + x³ – 8x² + 5x + 1\\ \underline{\begin{matrix} &&&&3 &1 &&2&-6&-1\end{matrix} } \\ \begin{matrix}3 &1 &2&-6&-1&0\end{matrix} &← ƒ(1) =0\end{matrix}
Because the remainder is 0, x – 1 is a factor. Additionally, we can determine from the coefficients in the bottom row that the other factor is
3x^{4} + 1x³+ 2x² – 6x – 1.
Thus, we can express the polynomial in factored form.
ƒ(x)= (x – 1)(3x^{4} + x³ + 2x² – 6x – 1)