If the polynomial $y^{3} - 5 y^{2} + 7 y + m$ is divided by y + 2, it leaves no remainder. The find the value of $m$ .

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Solution

If the polynomial $y^{3} - 5 y^{2} + 7 y + m$ is divided by $y + 2$ and leaves no remainder then it means that $y + 2$ is the factor of the given polynomial. So, by remainder theorem we substitute $y = - 2$ in $y^{3} - 5 y^{2} + 7 y + m$ and equate it to zero to find the value of $m$ as shown below:

$y^{3} - 5 y^{2} + 7 y + m = 0 \Rightarrow (- 2)^{3} - 5 (- 2)^{2} + (7 \times - 2) + m = 0 \Rightarrow - 8 - (5 \times 4) - 14 + m = 0 \Rightarrow - 8 - 20 - 14 + m = 0 \Rightarrow - 42 + m = 0 \Rightarrow m = 42$

Hence, $m = 42$ .

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If the polynomial y3−5y2+7y+m is divided by y + 2, it leaves no remainder. The find the value of m.

If the polynomial $y^{3} - 5 y^{2} + 7 y + m$ is divided by y + 2, it leaves no remainder. The find the value of $m$ .