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Worked solution

Read polynomial roots as intersection inputs

The curve CC has equation y=x2−2x+2x+1y=\dfrac{x^2-2x+2}{x+1}.

(i) Determine the asymptotes of CC.

(ii) Sketch CC, showing clearly any axial intercepts, turning points and asymptotes.

(iii) Deduce the number of real roots of the equation x4−2x3+2x2−x−1=0x^4-2x^3+2x^2-x-1=0.

CJC · Lecture · Example 12

Answer

(i) x=−1x=-1 and y=x−3y=x-3. (ii) Sole axial intercept (0,2)(0,2); local maximum (−1−5,−4−25)(-1-\sqrt5,-4-2\sqrt5) and local minimum (−1+5,−4+25)(-1+\sqrt5,-4+2\sqrt5). (iii) Two real roots.

(i) Preserve the division and remainder

Answer

x=−1x=-1, y=x−3y=x-3.

Both copies show quotient x−3x-3: subtract x(x+1)=x2+xx(x+1)=x^2+x from the numerator to leave −3x+2-3x+2, then subtract −3(x+1)=−3x−3-3(x+1)=-3x-3 to leave 5. Thus

y=x2−2x+2x+1=x−3+5x+1.\begin{aligned}&y\\ & =\frac{x^2-2x+2}{x+1}\\ & =x-3+\frac5{x+1}.\end{aligned}

The source asymptotes are x=−1x=-1 and y=x−3y=x-3. Independently the numerator at −1 is 5, so the vertical asymptote is genuine.

How did this part go?

(ii) Match exact features with the printed rounded coordinates

Both sketches label (0,2)(0,2), local maximum (−3.24,−8.47)(-3.24,-8.47) and local minimum (1.24,0.472)(1.24,0.472). Independently the numerator (x−1)2+1>0(x-1)^2+1>0 has no real zero, so there is no xx-intercept. Differentiating the division form gives

y′=1−5(x+1)2,x=−1±5,y=−4±25.y'=1-\frac5{(x+1)^2},\qquad x=-1\pm\sqrt5,\qquad y=-4\pm2\sqrt5.

The minus pair is the left local maximum; the plus pair is the right local minimum. Their exact values agree with the rounded source labels. The y=0y=0 label in the combined source picture belongs to the axis/reference curve; it is not a horizontal asymptote of CC.

How did this part go?

(iii) Preserve the algebra linking the two graphs

Answer

Two real roots.

The Teacher completes the blank Student working with

x4−2x3+2x2−x−1=0,x4−2x3+2x2=x+1,x2(x2−2x+2)=x+1,x2−2x+2x+1=1x2.\begin{aligned}x^4-2x^3+2x^2-x-1&=0,\\x^4-2x^3+2x^2&=x+1,\\x^2(x^2-2x+2)&=x+1,\\\frac{x^2-2x+2}{x+1}&=\frac1{x^2}.\end{aligned}

Sketch y=1/x2y=1/x^2 on the same axes. The Teacher adds that blue curve and marks its two intersections with CC, giving two real roots. Its lecturer note stresses that the x-coordinates of the intersection points solve the original equation; the plotted ordinates are not the roots.

Independently the rearrangement excludes x=0,−1x=0,-1, but neither solves the quartic: its values there are −1 and 5. Thus no original root is lost and no excluded input is promoted.

Expand diagram

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Curve C with pole −1, slant x−3, intercept (0,2) and rounded turns (−3.24,−8.47),(1.24,0.472). Reference 1/x² has pole 0 and horizontal 0; two circled intersections have x-coordinates equal to quartic roots.
Curve C with pole −1, slant x−3, intercept (0,2) and rounded turns (−3.24,−8.47),(1.24,0.472). Reference 1/x² has pole 0 and horizontal 0; two circled intersections have x-coordinates equal to quartic roots.
Curve C with pole −1, slant x−3, intercept (0,2) and rounded turns (−3.24,−8.47),(1.24,0.472). Reference 1/x² has pole 0 and horizontal 0; two circled intersections have x-coordinates equal to quartic roots.
How did this part go?

Verify that the sketch has not missed further roots

Independent check: let Q(x)=x4−2x3+2x2−x−1Q(x)=x^4-2x^3+2x^2-x-1. Then

Q′′(x)=12x2−12x+4=12(x−1/2)2+1>0.\begin{aligned}&Q''(x)\\ & =12x^2-12x+4\\ & =12(x-1/2)^2+1>0.\end{aligned}

Its derivative is strictly increasing, so QQ decreases then increases and has at most two real zeros. Since Q(−1)=5Q(-1)=5, Q(0)=−1Q(0)=-1 and Q(2)=5Q(2)=5, continuity gives one zero in (−1,0)(-1,0) and one in (0,2)(0,2). These are exactly the two graph intersections. This verifies the source count without treating a finite screen as proof.

How did this part go?

Common mistakes

Intersection ordinates are not polynomial roots. Distinguish the reference curve’s asymptote y=0 from C’s slant asymptote.

Solution notes

Teacher completes part (iii), adds reference curve/intersection marks and the x-coordinate teaching note. All source algebra, long division and rounded graph labels retained. No printed marks or external attribution.

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