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...