Question

check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial: x^2+3x+1,3x^4+5x^3-7x2+2x+2​

Answer 1

Question :-

check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial: x^2+3x+1,3x^4+5x^3-7x^2 + 2x + 2

Step-by-step explanation:

Step 1 :- Divide x^2 + 3x + 1 by 3x^4 + 5x^3 - 7x^2 + 2x + 2

Step 2 :- After dividing the given polynomial we get remainder as 0 and Quotient as 3x^2 - 4x + 2

Step 3 :- Therefore, x^2 + 3x + 1 is the factor of the given polynomial 3x^4+5x^3-7x^2 + 2x + 2.

Additional Information :-

$\mathrm{ \star \: ax^{2} + bx + c = 0}$

$\star \mathrm{\: ax^{3} + bx^{2} + cx + d = 0}$

$\star \: \mathrm{\alpha + \beta = \frac{ - b}{a}}$

$\mathrm{ \star \: \alpha \beta = \frac{c}{a}} \\ \mathrm{\star \: \alpha\beta +\beta\gamma + \gamma \alpha = \frac{c}{a}}\\ \mathrm{ \star \:\alpha\beta \gamma = \frac{ - d}{a}}\\ \star \mathrm{\alpha + \beta + \gamma = \frac{ - b}{a}}$

Answer 2

Explanation:-

Let the first polynomial be g(x) and other be p(x) .

• When we divide the p(x) by g(x) if the remainder become 0 .Then it is the factor of given polynomial .

• Let divide the polynomial 3x⁴ + 5x³ -7x²+ 2x +2 by polynomial x² + 3x + 1.

• Now we have to change the sign + → - and - → + in case of division.

• When remainder becomes 0 it is factor of given polynomial .

• If remainder is not zero it is not factor of given polynomial.

But as we can see in attachment the remainder becomes 0.

Hence, the x² + 3x + 1 is factor of given polynomial 3x⁴ + 5x³ -7x² +2x +2.