Zadanie 1a
Take b=2,d=3. (i) Sketch y=f(x), labelling its axis intercepts. (ii) Hence explain why f(x)=0 has no real root.
Polynomial factors and root relations
Use factor and remainder arguments, root sums and products, and graph behaviour to analyse polynomial equations.
Tę umiejętność sprawdza 60 zadań wzorowanych na arkuszach IB z lat 2022–2026.
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Zadanie 1b
Take b=−2,d=1. Sketch y=f(x), labelling its axis intercepts.
Zadanie 1d
For a real double root α, describe the identifying feature of y=f(x) at x=α.
Zadanie 1e
Prove that −α is also a double root.
Zadanie 1f
(i) Using f′(α)=0 or otherwise, show that α4 = −b/2. (ii) Hence show that d = b2/4.
Zadanie 1g
Under the double-root assumption, state the b-values for which f(x)=0 has (i) at least one real root; (ii) at least one purely imaginary root.
Zadanie 4a
Find the other two roots.
- Materiał graficzny
Zadanie 8a
Using sums and products of roots: (i) show that a=3; (ii) find b.
Zadanie 8b
Express P as (Az + B)(z2 + Cz + D)(z2 + Ez + F) with integer coefficients.
Zadanie 12a
First suppose is finite. (i) Show that b=2a−4. (ii) Find the limit in terms of a.
Zadanie 12d
Verify that z²−9 is a factor of P(z).
Zadanie 12e
The already-specified quadratic factor is z2 − 9. P is the product of three quadratic polynomials with real coefficients. Find the other two quadratic factors.
Zadanie 2
Let F(x) = 3x4 − 4x3 + px2 + qx + 1. The polynomial has factor x + 1, and division by x − 2 leaves remainder 45. Determine p and q.
Zadanie 5b
Find the allowable values of k if the two solutions also have the same sign.
Zadanie 7
The polynomial F(x) = x4 + px3 − 2x2 + qx − 8 is divisible by (x − 2)2. Determine the real coefficients p and q.
Zadanie 12d
These four roots obey . Their reciprocals obey , where p, q, r are integers with no common factor and p > 0. Find p, q, r.
Zadanie 12d
The monic quartic is z4 + az3 + bz2 + cz + d = 0, with real coefficients. Two distinct roots z1, z2 satisfy the circle condition and . Find the coefficient a of this…
Zadanie 1e
The normal to F at A is y = −t2x + kt3 − k/t. (i) Show that its intersections with F satisfy x2 − k(t − t−3)x − k2/t2 = 0. (ii) Hence, using a sum/product of roots or another valid method, find B.
Zadanie 6
Let P(x) = 3x2 + qx + r, with real coefficients and real roots α,β. Division by x+1 leaves remainder 14. Given α2 + β2 = 23/3, find q and r.
Zadanie 2h
For F, show that .
Zadanie 2i
For F, find the exact value of (xB − xA)/(xC − xB).
Zadanie 7a
State α + β + γ.
Zadanie 7b
Show that p=2.
Zadanie 7c
It is further given that h(−1/3) = 0 and α, β are positive integers with α < β. (i) Show that αβ = 3. (ii) Find α and β.
Zadanie 10f
Find the product of the solutions of f(x)=g(x).
Zadanie 11a
Show that x−1 is a factor of P.
Zadanie 11b
Hence express P as a product of three linear factors.
Zadanie 2c
For p(x) = ax2 + bx + a, where a≠0, verify p(x) = x2p(1/x) for x≠0.
Zadanie 2d
If α≠0 is a root of a palindromic polynomial p of degree n, prove that 1/α is also a root.
Zadanie 2e
Prove that the product of two palindromic polynomials is palindromic.
Zadanie 2f
Write F(x) = (x4 + ux2 + 1)(x4 + vx2 + 1), where u,v are integers with u<v. Determine u,v by forming and solving an appropriate system of equations.
Zadanie 2g
Hence find all roots of F(x) = 0 exactly, stating multiplicities.
Zadanie 2h
Prove that every monic palindromic polynomial of odd degree has −1 as a root.
Zadanie 2d
For the cubic family, state (i)r1 + r2 and (ii)r1r2 in terms of a,b,c.
Zadanie 2e
For an admissible cubic F: (i) express r2 − r1 in terms of a,b; (ii) use the sum and difference of the stationary abscissas to show that r1 = (3a2 − 3ab − 2b)/(6a); (iii) use this result to show that b=0 or a=−1/3.
Zadanie 2f
Determine the two admissible cubic functions with b=0.
Zadanie 2g
Now set a=−1/3. (i) Express r₁ in terms of b. (ii) Hence or otherwise find the exact pairs (b,c) making F admissible. Give each value as a rational expression involving a square root.
Zadanie 7
For m>0, let P(z) = 5m − mz + (20/m)z2 − z3. Given that z−2i is a factor, find every root of P(z)=0.
Zadanie 11c
(i) Find the other roots of p(z)=0. (ii) By a suitable transformation or otherwise, solve 1+3w−6w²+20w³=0.
Zadanie 12f
The inflexion point P lies in . Given y′′ = (−4x4 + 2x2y2 + 2y4)/(x2y3), use this expression to show that P lies on a line y=mx and determine m exactly.
Zadanie 1a
Set a=−4. By varying b systematically or otherwise, find the two b values for which F has a repeated real root.
Zadanie 1b
For a=−4, state the b values giving (i) no x-axis intercepts; (ii) four distinct x-axis intercepts.
Zadanie 1c
Now set a=4. State the b values giving (i) exactly one intercept; (ii) no intercepts; (iii) exactly two intercepts.
Zadanie 1h
Prove that a<0 and 0<b<a²/4 imply exactly four distinct x-axis intercepts.
Zadanie 1f
A rational point has two rational coordinates. On C : y2 = x3 − 3x + 4, let P=(0,2). The tangent at P meets C again at Q; let S=(0,−2). (i) Find the tangent equation at P. (ii) Hence find Q, expressing its coordinates as fractions.
Zadanie 1g
A rational point has two rational coordinates. On C : y2 = x3 − 3x + 4, let P=(0,2). The tangent at P meets C again at Q; let S=(0,−2). Use the tangent construction to determine Q, then find the third intersection of the full straight line through Q and S with C.
Zadanie 2g
Let g(x) = (x − r)2[(x − a)2 + b2], b>0, with A=(a,g(a)) and R=(r,0). When a≠r the tangent at A has x-intercept T=((a+r)/2,0). Assume (a − r)2 > 2b2, so the quartic has two inflection points. (i) Show that their x-coordinates satisfy…
Zadanie 2a
Expand (z−u)(z−v)(z−w) and show that u+v+w=−q/r, uv+vw+wu=p/r, and uvw=−1/r.
Zadanie 2b
You may use u+v+w=−q/r and uv+vw+wu=p/r. (i) Show that u2 + v2 + w2 = (q2 − 2pr)/r2. (ii) Hence show that (u − v)2 + (v − w)2 + (w − u)2 = 2(q2 − 3pr)/r2.
Zadanie 2d
Consider Fc(x) = x3 + cx2 − 6x + 1, where c is real. You may use the sufficient condition: a real cubic x³+px²+qx+r, r≠0, has a nonreal root if q²<3pr. Show that F₁₃(x)=0 has a nonreal root.
Zadanie 2f
A real quartic x4 + px3 + qx2 + rx + s = 0, s≠0, has nonzero roots α,β,γ,δ. Their reciprocals u,v,w,z satisfy u+v+w+z=−r/s and uv+uw+uz+vw+vz+wz=q/s. (i) Express u²+v²+w²+z² in terms of q,r,s. (ii) Hence state a sufficient inequality in q,r,s for the original…
Zadanie 2g
A real quartic with coefficients q,r,s as in x⁴+px³+qx²+rx+s has a nonreal root whenever r²<2qs. Use this to show that x4 + 2x3 + 4x2 − 3x + 2 = 0 has a nonreal root.
Zadanie 2h
Consider x4 + 8x3 − 15x2 − 9x − 2 = 0, which has exactly one integer root. You may use two sufficient criteria: a quartic x⁴+px³+qx²+rx+s has a nonreal root if r²<2qs; a cubic x³+px²+qx+r, r≠0, has one if q²<3pr. (i) State what the quartic criterion tells you here.…
Zadanie 8a
Express the product of its roots in terms of k.
Zadanie 8b
Determine k for which one root is positive and the other negative.
Zadanie 12d
For this 120° configuration, suppose z₁,z₂ are the distinct roots of z2 + az + b = 0 with real a,b. Use z12 + z22 = −z1z2 to show that a²−b=0.
Zadanie 12e
Now consider z2 + az + 16 = 0. You may use a²=b for the 120° configuration. Deduce that the argument restriction 0<α−θ<π permits only one such triangle.
Zadanie 5a
Use the product of the four roots to find a.
Zadanie 5b
Find p.
Zadanie 11b
Suppose x2 − 3x + 2 is a factor of P. (i) Express b in terms of a. (ii) Hence show that a=5,b=8,c=4 is the only permitted assignment. (iii) Factor P completely into linear factors. (iv) Sketch y=P(x), labelling all intercepts.