Polynomials Set-2

Q1. A quadratic polynomial $p(x)=x^2+px+q$ satisfies $p(1)=6$ and $p(2)=11$. What is the value of $p+q$?

Q2. A quadratic polynomial with integer coefficients has roots $r$ and $3r$. If the sum of its coefficients is zero, then the possible values of $r$ are:

Q3. Let $p(x)$ be a polynomial of degree at most $3$ such that $p(0)=0,\; p(1)=10,\; p(2)=20,\; p(3)=30$. Find $p(4)$.

Q4. Find all real values of $a$ for which the polynomial $x^{4}+ax^{3}+(a-2)x^{2}-ax+1$ is divisible by $x^{2}+1$.

Q5. Let $p(x)$ be a polynomial with integer coefficients such that $p(0)=0$ and $p(1)=p(2)=p(3)=2016$. What is the minimal possible degree of $p(x)$?

Q6. For the polynomial $p(x)=x^2+kx+6$, it is given that $p(1)=p(-2)$. What is the value of $k$?

Q7. A polynomial $f(x)$ leaves remainder $3$ when divided by $x-2$ and remainder $5$ when divided by $x-3$. What is the remainder when $f(x)$ is divided by $(x-2)(x-3)$?

Q8. Let $p(x)=x^3-6x^2+px+q$. If $(x-1)$ is a factor of $p(x)$ and the remainder when $p(x)$ is divided by $(x-2)$ is $3$, determine the pair $(p,q)$.

Q9. Let $p(x)$ be a monic cubic polynomial (leading coefficient $1$) such that the remainders are $p(1)=2,\; p(2)=3,\; p(3)=4$ when divided respectively by $x-1,\;x-2,\;x-3$. What is $p(0)$?

Q10. If $f(x)=x^4+ax^3+bx^2+cx+d$ satisfies $f(1)=f(-1)=f(2)=f(-2)=0$, what is the value of $a+b+c+d$?

Q11. If the polynomial $p(x)=2x^3-3x^2+4x-5$ is divided by $x-2$, what is the remainder?

Q12. Find the values of $a$ and $b$ such that the polynomial $2x^3+x^2-ax+b$ is exactly divisible by $(x-1)(x+2)$.

Q13. For the polynomial $p(x)=x^3+(k-2)x^2+(k+1)x-4$, determine the value(s) of $k$ for which $(x-1)^2$ is a factor of $p(x)$.

Q14. Let

For which real value(s) of $k$ do $p(x)$ and $q(x)$ have a common linear factor?

Q15. The cubic $f(x)=x^3-3x+k$ has a repeated (double) real root for which value(s) of $k$?