A few polynomial tasks

[Johulus]

1.

When polynomial $\displaystyle f(x)$ is being divided by polynomial $\displaystyle g_{1}(x)=2x - 1$, the remainder is 2. When $\displaystyle f(x)$ is divided by $\displaystyle g_{2}(x)= x + 2$, the remainder is 3. Determine the remainder $\displaystyle r$ which you get when $\displaystyle f(x)$ is divided by $\displaystyle g_{1}(x)\cdot g_{2}(x)$ .

2.

Determine the polynomial of a second degree which when divided by $\displaystyle g_{1}(x)= x+1$ gives the remainder 11, when divided by $\displaystyle g_{2}(x)= x + 2$ gives the remainder 17 and when divided by $\displaystyle g_{3}(x)=x$ gives the remainder 7.

3.

Polynomials $\displaystyle f_{1}$ and $\displaystyle f_{2}$, when divided by polynomial $\displaystyle g$, give the remainders $\displaystyle r_{1}(x)=x^2-3\cdot x + 1$ and $\displaystyle r_{2}(x) = 2\cdot x - 1$. Determine the remainder when the polynomial $\displaystyle 2\cdot f_{1}(x) + 3\cdot f_{2}(x)$ is divided by $\displaystyle g(x)$ .

I can't get the hang of these. I am not sure what should I do here. Any help is welcome.

 

[Johulus]

I have to say that I have successfully solved the third task, but would still appreciate some hint for the first two.

 

[Deleted member 4993]

1.

When polynomial $\displaystyle f(x)$ is being divided by polynomial $\displaystyle g_{1}(x)=2x - 1$, the remainder is 2. When $\displaystyle f(x)$ is divided by $\displaystyle g_{2}(x)= x + 2$, the remainder is 3. Determine the remainder $\displaystyle r$ which you get when $\displaystyle f(x)$ is divided by $\displaystyle g_{1}(x)\cdot g_{2}(x)$ .

2.

Determine the polynomial of a second degree which when divided by $\displaystyle g_{1}(x)= x+1$ gives the remainder 11, when divided by $\displaystyle g_{2}(x)= x + 2$ gives the remainder 17 and when divided by $\displaystyle g_{3}(x)=x$ gives the remainder 7.

3.

Polynomials $\displaystyle f_{1}$ and $\displaystyle f_{2}$, when divided by polynomial $\displaystyle g$, give the remainders $\displaystyle r_{1}(x)=x^2-3\cdot x + 1$ and $\displaystyle r_{2}(x) = 2\cdot x - 1$. Determine the remainder when the polynomial $\displaystyle 2\cdot f_{1}(x) + 3\cdot f_{2}(x)$ is divided by $\displaystyle g(x)$ .

I can't get the hang of these. I am not sure what should I do here. Any help is welcome.

What are your thoughts?

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[Johulus]

Concerning the second task, I've managed to solve it....

$\displaystyle f(x)-11=(x+1)\cdot q_{1}(x) \qquad f(x)-17=(x+2)q_{2} \qquad f(x)-7=x\cdot q_{3}$
Since the polynomial is of the second degree it's in the form $\displaystyle f(x)=ax^2+bx+c$.
So, what I did is $\displaystyle \dfrac{f(x)-11}{x+1}=q_{1} ... \dfrac{ax^2+bx+c-11}{x+1}=...$ I've divided them so the leading coefficient of the remainder is of the smaller degree than the divisor. After I've done it with all three. I've just taken the expressions for remainders from all three and equalised them to 0 since the remainder in all these 3 cases must be 0. I've got 3 linear equations with 3 variables.

Concerning the first task, I am still stuck....

I did the following...

$\displaystyle f(x)=(2x-1)\cdot q_{1} + 2 \qquad f(x)=(x+2)\cdot q_{2} +3 \\ f(x)-2=(2x-1)\cdot q_{1} \qquad f(x)-3=(x+2)\cdot q_{2} \\ \dfrac{(f(x)-2)(f(x)-3)}{(2x-1)(x+2)}=q_{1}\cdot q_{2}$
And that's where I am stuck.