Question

Find whether or not the first polynomial is a factor of the second polynomial.

3x+2, 3x⁴+5x³-x²+13x+10​

Answer 1

Step-by-step explanation:

$3 {x}^{4} + 5 {x}^{3} - {x}^{2} + 13x + 10 \\ 3x + 2 = 0 \\ 3x = - 2 \\ x = - \frac{2}{3} \\ \\ 3 {( \frac{ - 2}{3} )}^{4} + 5 {( \frac{ - 2}{3} )}^{3} - {( \frac{ - 2}{3} )}^{2} + 13( \frac{ - 2}{3} ) + 10 \\ = 3 \times \frac{ 16}{81} + 5 \times \frac{ - 8}{27} - \frac{4}{9} - \frac{26}{3} + 10 \\ = \frac{16}{27} - \frac{40}{27} - \frac{4}{9} - \frac{26}{3} + 10 \\ = \frac{16 - 40}{27} - \frac{4}{9} - \frac{26}{3} + 10 \\ = - \frac{24}{27} - \frac{4}{9} - \frac{26}{3} + 10 \\ = - \frac{8}{9} - \frac{4}{9} - \frac{26}{3} + 10 \\ = \frac{ - 8 - 4}{9} - \frac{26}{3} + 10 \\ = - \frac{ 12}{9} - \frac{26}{3} + 10 \\ = \frac{ - 12- 78 + 90}{9} \\ = \frac{ - 90 + 90}{9} \\ = \frac{0}{9} \\ = 0$

3x + 2 is the factor of the polynomial

Hope you get your answer