CBSE Class 9 Answered

Examine if (x + 1), (x + 2) and (x - 3) are factors of the polynomial x³ - 2x² - 5x + 6.

Asked by Topperlearning User | 14 Sep, 2017, 02:21: PM

Expert Answer

$straight p left parenthesis straight x right parenthesis space equals space straight x cubed space minus space 2 straight x squared space minus space 5 straight x space plus space 6 left parenthesis straight x space plus space 1 right parenthesis space is space straight a space factor space of space straight p left parenthesis straight x right parenthesis space if space straight p left parenthesis negative 1 right parenthesis space equals space 0 Now comma space straight p left parenthesis negative 1 right parenthesis space equals space left parenthesis negative 1 right parenthesis cubed space minus space 2 left parenthesis negative 1 right parenthesis squared space minus space 5 left parenthesis negative 1 right parenthesis space plus space 6 space equals space minus 1 space minus space 2 space plus space 5 space plus space 6 space equals space 8 space not equal to space 0 Hence comma space left parenthesis straight x space plus space 1 right parenthesis space is space not space straight a space factor space of space straight p left parenthesis straight x right parenthesis. left parenthesis straight x space plus space 2 right parenthesis space is space straight a space factor space of space straight p left parenthesis straight x right parenthesis space if space straight p left parenthesis negative 2 right parenthesis space equals space 0 Now comma space straight p left parenthesis negative 2 right parenthesis space equals space left parenthesis negative 2 right parenthesis cubed space minus space 2 left parenthesis negative 2 right parenthesis squared space minus space 5 left parenthesis negative 2 right parenthesis space plus space 6 space equals space minus 8 space minus space 8 space plus space 10 space plus space 6 space equals space 0 Hence comma space left parenthesis straight x space plus space 2 right parenthesis space is space straight a space factor space of space straight p left parenthesis straight x right parenthesis. left parenthesis straight x space minus space 3 right parenthesis space is space straight a space factor space of space straight p left parenthesis straight x right parenthesis space if space straight p left parenthesis 3 right parenthesis space equals space 0 Now comma space straight p left parenthesis 3 right parenthesis space equals space left parenthesis 3 right parenthesis cubed space minus space 2 left parenthesis 3 right parenthesis squared space minus space 5 left parenthesis 3 right parenthesis space plus space 6 space equals space 27 space minus space 18 space minus space 15 space plus space 6 space equals space 0 Hence comma space left parenthesis straight x space minus space 3 right parenthesis space is space straight a space factor space of space straight p left parenthesis straight x right parenthesis.$