IB Maths AA HL Polynomials and Rational Functions Questions

1. State the Remainder Theorem: If a polynomial $P(x)$ is divided by $(x-a)$, the remainder is $P(a)$. Here, $a=2$.

2. Set up the equation using the given remainder: $P(2)=10$.

3. Substitute $x=2$ into the polynomial: $2(2{)}^3-(2{)}^2+k(2)-4=10$.

4. Evaluate the powers first: $2(8)-4+2k-4=10$.

5. Simplify the constants: $16-8+2k=10⟹8+2k=10$.

6. Solve for $k$: $2k=2⟹\mathbf{k}\mathbf{=}\mathbf{1}$.

1. Recognize that an oblique asymptote exists because the degree of the numerator (2) is exactly one higher than the degree of the denominator (1).

2. Perform algebraic long division or synthetic division. Divide $x^2+5x-2$ by $x+2$.

3. Calculate the first term: $x^2÷x=x$. Multiply $x(x+2)=x^2+2x$ and subtract to leave $3x-2$.

4. Calculate the second term: $3x÷x=3$. Multiply $3(x+2)=3x+6$ and subtract to leave a remainder of $-8$.

5. Rewrite the function: $f(x)=x+3-\frac{8}{x+2}$.

6. Conclude: As $x\to \pm \mathrm{\infty }$, the fractional remainder $\frac{8}{x+2}\to 0$. The oblique asymptote is the linear quotient, $\mathbf{y}\mathbf{=}\mathbf{x}\mathbf{+}\mathbf{3}$.

1. Identify the standard properties of roots for a cubic $a{x}^3+b{x}^2+cx+d=0$. The sum of the roots is $\sum \alpha =\alpha +\beta +\gamma =-\frac{b}{a}$.

2. Calculate the sum: $\alpha +\beta +\gamma =-\frac{-4}{2}=2$.

3. Identify the sum of the product of pairs: $\sum \alpha \beta =\alpha \beta +\beta \gamma +\gamma \alpha =\frac{c}{a}=\frac{5}{2}$.

4. Recall the algebraic expansion identity for a trinomial squared: $(\alpha +\beta +\gamma {)}^2={\alpha }^2+{\beta }^2+{\gamma }^2+2(\alpha \beta +\beta \gamma +\gamma \alpha )$.

5. Substitute the known values into the identity: $2^2={\alpha }^2+{\beta }^2+{\gamma }^2+2\left(\frac{5}{2}\right)$.

6. Simplify and solve for the sum of squares: $4={\alpha }^2+{\beta }^2+{\gamma }^2+5⟹{\alpha }^{\mathbf{2}}\mathbf{+}{\beta }^{\mathbf{2}}\mathbf{+}{\gamma }^{\mathbf{2}}\mathbf{=}\mathbf{-}\mathbf{1}$.