DP Mathematics HL Questionbank

• Syllabus

2.5

Sub sections and their related questions

Polynomial functions and their graphs.

• 12M.1.hl.TZ2.1: The same remainder is found when $2{x^3} + k{x^2} + 6x + 32$ and...
• 12M.1.hl.TZ2.12B.a: Explain why, of the four roots of the equation $f(x) = 0$ , two are real and two are complex.
• 13M.1.hl.TZ1.12a: Express $4{x^2} - 4x + 5$ in the form $a{(x - h)^2} + k$ where a, h, $k \in \mathbb{Q}$.
• 13M.2.hl.TZ1.6: A polynomial $p(x)$ with real coefficients is of degree five. The equation $p(x) = 0$ has a...
• 16M.2.hl.TZ1.8: When ${x^2} + 4x - b$ is divided by $x - a$ the remainder is 2. Given that...
• 17N.1.hl.TZ0.3a: Given that $q(x)$ has a factor $(x - 4)$, find the value of $k$.
• 17N.1.hl.TZ0.3b: Hence or otherwise, factorize $q(x)$ as a product of linear factors.
• 18M.1.hl.TZ1.1: Let f(x) = x4 + px3 + qx + 5 where p, q are constants. The remainder when f(x) is divided by (x...
• 18M.2.hl.TZ2.2: The polynomial ${x^4} + p{x^3} + q{x^2} + rx + 6$ is exactly divisible by each...

The factor and remainder theorems.

• 10M.1.hl.TZ1.1: Given that $A{x^3} + B{x^2} + x + 6$ is exactly divisible by $(x +1)(x − 2)$, find the value...
• 12M.1.hl.TZ2.1: The same remainder is found when $2{x^3} + k{x^2} + 6x + 32$ and...
• 12M.1.hl.TZ2.12B.a: Explain why, of the four roots of the equation $f(x) = 0$ , two are real and two are complex.
• 12M.1.hl.TZ2.12B.b: The curve passes through the point $( - 1,\, - 18)$ . Find $f(x)$ in the...
• 08M.1.hl.TZ2.2: The polynomial $P(x) = {x^3} + a{x^2} + bx + 2$ is divisible by (x +1) and by (x − 2) . Find...
• 08N.1.hl.TZ0.1: When $f(x) = {x^4} + 3{x^3} + p{x^2} - 2x + q$ is divided by (x − 2) the remainder is 15, and...
• 11M.1.hl.TZ2.12a: Factorize ${z^3} + 1$ into a linear and quadratic factor.
• 09M.1.hl.TZ2.1: When the function $q(x) = {x^3} + k{x^2} - 7x + 3$ is divided by (x + 1) the remainder is seven...
• 09N.1.hl.TZ0.1: When $3{x^5} - ax + b$ is divided by x −1 and x +1 the remainders are equal. Given that a ,...
• SPNone.2.hl.TZ0.1: Given that (x − 2) is a factor of $f(x) = {x^3} + a{x^2} + bx - 4$ and that division $f(x)$...
• 11M.2.hl.TZ1.4a: Find the value of $a$ .
• 14M.1.hl.TZ1.1: When the polynomial $3{x^3} + ax + b$ is divided by $(x - 2)$, the remainder is 2, and when...
• 13N.1.hl.TZ0.1: The cubic polynomial $3{x^3} + p{x^2} + qx - 2$ has a factor $(x + 2)$ and leaves a remainder...
• 14N.2.hl.TZ0.6: Consider $p(x) = 3{x^3} + ax + 5a,\;\;\;a \in \mathbb{R}$. The polynomial $p(x)$ leaves a...
• 16M.2.hl.TZ1.8: When ${x^2} + 4x - b$ is divided by $x - a$ the remainder is 2. Given that...
• 17N.1.hl.TZ0.3a: Given that $q(x)$ has a factor $(x - 4)$, find the value of $k$.
• 17N.1.hl.TZ0.3b: Hence or otherwise, factorize $q(x)$ as a product of linear factors.
• 18M.1.hl.TZ1.1: Let f(x) = x4 + px3 + qx + 5 where p, q are constants. The remainder when f(x) is divided by (x...
• 18M.2.hl.TZ2.2: The polynomial ${x^4} + p{x^3} + q{x^2} + rx + 6$ is exactly divisible by each...

The fundamental theorem of algebra.

• 12M.1.hl.TZ2.1: The same remainder is found when $2{x^3} + k{x^2} + 6x + 32$ and...
• 12M.1.hl.TZ2.12B.a: Explain why, of the four roots of the equation $f(x) = 0$ , two are real and two are complex.
• 12M.1.hl.TZ2.12B.b: The curve passes through the point $( - 1,\, - 18)$ . Find $f(x)$ in the...
• 11M.1.hl.TZ2.12a: Factorize ${z^3} + 1$ into a linear and quadratic factor.
• 13M.1.hl.TZ2.13b: (i)     State the solutions of the equation ${z^7} = 1$ for $z \in \mathbb{C}$, giving them...
• 14M.2.hl.TZ1.1: One root of the equation ${x^2} + ax + b = 0$ is $2 + 3{\text{i}}$ where...