Choose a representation
Use substitution to find the remainder on division by x-c. For ax²+bx+c, the sum of roots is -b/a; a repeated real root requires b²-4ac=0.
Worked example: follow the quantity
The graph of R(x)=(x²−9)/(x−3) has a removable hole: one point missing from an otherwise continuous line. What is the y-coordinate of that missing point? The original denominator is zero at x=3; keep that input excluded when factoring. For permitted inputs, cancel (x−3) to obtain the line y=x+3. Evaluate this extended line at the excluded input to locate the hole height. R(x)=x+3 only when x≠3; the extended line has y=3+3=6 at the missing input. R(3) itself stays undefined. Therefore the requested value is 6.
Contrast a plausible mistake
For that example, the tempting answer 0 follows this error: Treat the undefined quotient 0/0 as zero instead of using the extended line. The requested quantity is 6. Explain why the wrong operation changes the mathematical object, not merely the arithmetic.
A second representation
The two roots of 2 x²−6 x+5=0 are r and s, counted with multiplicity. What is r+s? For ax²+bx+c, the root sum is −b/a, including complex roots. Divide the negated linear coefficient by the leading coefficient. Substitute the given numbers: 6 ÷ 2 = 3. Therefore the requested value is 3.
Transfer before checking
The graph of R(x)=(x²−16)/(x−4) has a removable hole: one point missing from an otherwise continuous line. What is the y-coordinate of that missing point? Try without viewing the answer. Check: 8. The original denominator is zero at x=4; keep that input excluded when factoring. For permitted inputs, cancel (x−4) to obtain the line y=x+4. Evaluate this extended line at the excluded input to locate the hole height. R(x)=x+4 only when x≠4; the extended line has y=4+4=8 at the missing input. R(4) itself stays undefined.
Scope and conditions
A missing point in a rational graph remains missing even when its formula simplifies. A repeated root is one distinct solution with multiplicity two, not two different crossings. These original practice tasks sample the official topic; they do not certify complete coverage or a predicted ACT score.
The graph of R(x)=(x²−9)/(x−3) has a removable hole: one point missing from an otherwise continuous line. What is the y-coordinate of that missing point?
- The original denominator is zero at x=3; keep that input excluded when factoring.
- For permitted inputs, cancel (x−3) to obtain the line y=x+3. Evaluate this extended line at the excluded input to locate the hole height.
- R(x)=x+3 only when x≠3; the extended line has y=3+3=6 at the missing input. R(3) itself stays undefined.
- Therefore the requested value is 6.
Compare the other answer choices
Treat the undefined quotient 0/0 as zero instead of using the extended line.
Report the excluded x-coordinate rather than the missing point’s y-coordinate.
Use the line y=x minus the positive root and evaluate it at the other numerator root.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
ACT Mathematics content