If (p(x)=x^2-2(b-4)x+(b-4)^2), at which (x)-value will the graph touch the (x)-axis?
Correct answer: A
It is ((x-(b-4))^2), so the repeated zero is (b-4). Tip: a perfect square form shows the zero quickly.
Mathematics
Geometrical meaning of the zeroes of a polynomial.
बहुपद के शून्यकों का ज्यामितीय अर्थ
In this Class 10 Mathematics topic from Polynomials, students understand the geometrical meaning of a polynomial’s zeroes by connecting algebraic expressions with their graphs. A zero is represented by the x-coordinate where the graph meets or touches the x-axis. Students learn how the number of points of intersection indicates the number of real zeroes, and interpret graphs of linear, quadratic, and other polynomial functions to relate their shapes and x-intercepts to solutions of p(x) = 0.
TOPIC PRACTICE
Quiz this set
Up to 25 questions from this page. Select your focus, then start.
25 questions
Choose questions
Expert · Level 3
View options
- (x=b-4)
- (x=b+4)
- (x=2b-8)
- (x=4-b)
Expert · Level 3
View options
- ((n,0)) and ((n+3,0))
- ((0,n)) and ((0,n+3))
- ((-n,0)) and ((-n-3,0))
- ((n,n+3)) and ((n+3,n))
Expert · Level 3
View options
- (11)
- (3)
- (15)
- (-11)
Expert · Level 3
View options
- Two, touches at both
- Two, crosses at both
- Eight, touches at all
- One, touches only at (x=-2)
Expert · Level 3
View options
- (c) and (-d)
- (-c) and (d)
- (c), (-d), (-d)
- Only (c)
Expert · Level 3
View options
- ((-4,0))
- ((-8,0))
- ((4,0))
- ((0,-4))
Expert · Level 3
View options
- (19)
- (25)
- (31)
- (-31)
Expert · Level 3
View options
- ((17,0)) and ((-4,0))
- ((-17,0)) and ((4,0))
- ((0,17)) and ((0,-4))
- ((68,0)) and ((-13,0))
Expert · Level 3
View options
- (0) and (-9)
- Only (-9)
- Only (0)
- (0) and (9)
Expert · Level 3
View options
- ((-10,0))
- ((10,0))
- ((20,0))
- ((0,100))
Expert · Level 3
View options
- (s+\frac{4}{3})
- (3s+4)
- (s-4)
- (s+7)
Expert · Level 3
View options
- Because it is ((x-7)^2+4)
- Because it is ((x-7)^2)
- Because its zeroes are (7) and (4)
- Because it is a constant polynomial
Expert · Level 3
View options
- The graph will cross the (x)-axis
- The graph will touch the (x)-axis
- The graph will not meet the (x)-axis
- (x=3) is not a zero
Expert · Level 3
View options
- (x=q-2)
- (x=q+2)
- (x=2q-4)
- (x=q-4)
Expert · Level 3
View options
- ((-12,0))
- ((5,0))
- ((0,14))
- ((14,0))
Expert · Level 3
View options
- Positive
- Negative
- Zero
- Cannot be determined
Expert · Level 3
View options
- Other (7), intersections ((6,0)), ((7,0))
- Other (-7), intersections ((6,0)), ((-7,0))
- Other (13), intersections ((6,0)), ((13,0))
- Other (0), intersections ((6,0)), ((0,0))
Expert · Level 3
View options
- Product (-144), sum (0)
- Product (144), sum (0)
- Product (0), sum (24)
- Product (-24), sum (144)
Expert · Level 3
View options
- \(\left(\frac{7}{6},0\right),\ \left(-\frac{7}{6},0\right)\)
- \((7,0),\ (-7,0)\)
- \(\left(\frac{6}{7},0\right),\ \left(-\frac{6}{7},0\right)\)
- No x-axis intersections
Expert · Level 3
View options
- One
- Two
- Three
- Four
Expert · Level 3
View options
- One
- Two
- Three
- Cannot be determined
Expert · Level 3
View options
- ((-6,0)) and ((6,0))
- ((-36,0)) and ((36,0))
- ((0,0)), ((-6,0)), ((6,0))
- None
Expert · Level 3
View options
- (6)
- (7)
- (8)
- (9)
Expert · Level 3
View options
- The graph cuts the (x)-axis twice
- The graph touches the (x)-axis once
- The graph does not cut the (x)-axis
- The graph is the (y)-axis
Expert · Level 3
View options
- (0,5,7)
- (0,-5,-7)
- (5,7,12)
- Only (0)
If (p(x)=x^2-(2n+3)x+n(n+3)), what will be the (x)-axis intersections of the graph?
Correct answer: A
The direct answer is option A: the intersections are \\(n,0\\) and \\(n+3,0\\). To find an x-axis intersection, put the y-value, here \\(p(x)\\), equal to zero. Factor the polynomial: \\(x^2-(2n+3)x+n(n+3)=(x-n)(x-(n+3))\\), because the roots add to \\(2n+3\\) and multiply to \\(n(n+3)\\). Thus \\(x=n\\) or \\(x=n+3\\). Points on the x-axis have y-coordinate zero, so the points are \\( (n,0)\\) and \\( (n+3,0)\\). Option A is correct. Option B gives points on the y-axis, because their x-coordinate is zero, so it does not show x-axis intersections. Option C uses negative roots, which do not come from the factorisation. Option D gives points whose y-coordinates are generally nonzero, so they are not on the x-axis. Memory cue: zero of a polynomial gives an x-coordinate, and always write the graph point as \\( (zero,0)\\).
If (p(-7)=0), (p(-3)<0), (p(4)=0), (p(8)>0), what is the distance between the given zeroes?
Correct answer: A
Direct answer: Option A, 11. A zero is identified by p(x)=0. The given equalities show p(-7)=0 and p(4)=0, so the zeroes are -7 and 4. The statements p(-3)<0 and p(8)>0 do not identify zeroes; they only tell us that the polynomial has negative and positive values at those inputs. The distance between two points on the number line is the absolute difference: |4-(-7)|=|4+7|=11. Option A is correct. Option B, 3, is only the distance from -7 to -3, not between the two zeroes. Option C, 15, is the distance from -7 to 8, not the requested pair. Option D, -11, cannot be a distance because distance is non-negative. Exam cue: first select only p(x)=0, then subtract and take the absolute value.
If (p(x)=(x+2)^6(x-5)^2), what are the number of distinct real zeroes and graph behavior?
Correct answer: A
There are two distinct zeroes (-2) and (5), and both have even powers. Tip: at an even-power zero the graph usually touches.
If (p(x)=(x-c)^5(x+d)^2), where (c\neq -d), what are the distinct zeroes?
Correct answer: A
The zeroes of a factored polynomial are found by setting each factor equal to zero. A factor raised to a power still contributes only one distinct zero; its exponent gives the multiplicity. This distinction is important because the question asks for distinct zeroes rather than all zeroes counted with repetition. The given inequality ensures that the two values do not coincide.
From \(x-c=0\), we obtain \(x=c\). From \(x+d=0\), we obtain \(x=-d\). The powers 5 and 2 mean that these zeroes have multiplicities 5 and 2, respectively, but the distinct list is only \(c\) and \(-d\). Since \(c\ne-d\), they are genuinely different. Therefore option A is correct; option C repeats \(-d\) unnecessarily.
The (x)-axis intersections of a parabola are ((-17,0)) and ((9,0)). What is the coordinate of the midpoint of its zeroes?
Correct answer: A
The midpoint is \(\left(\frac{-17+9}{2},0\right)=(-4,0)\). Tip: on the (x)-axis the midpoint has \(y=0\).
If a graph cuts the (x)-axis at (x=-18), (x=-6), (x=5), and (x=13), what is the range of the zeroes?
Correct answer: C
The range is the difference between the greatest and smallest zero, (13-(-18)=31). Tip: range is always non-negative.
If (p(x)=x^2-13x-68), what are the (x)-axis intersections of the graph?
Correct answer: A
(x^2-13x-68=(x-17)(x+4)), so the zeroes are (17) and (-4). Tip: write intersection points from factors.
If a graph cuts the (y)-axis at the origin and cuts the (x)-axis at (x=-9), what are the zeroes?
Correct answer: A
The origin is also on the (x)-axis, and (x=-9) is another (x)-axis intersection. Tip: count ((0,0)) as zero (0).
If (p(x)=x^2+20x+100), at which point will the graph touch the (x)-axis?
Correct answer: A
(x^2+20x+100=(x+10)^2), so the touching point is ((-10,0)). Tip: change the sign in a perfect square to get the zero.
If the (x)-axis intersections of a graph are ((s-4,0)), ((s+1,0)), ((s+7,0)), what is the mean of the zeroes?
Correct answer: A
The mean is (\frac{(s-4)+(s+1)+(s+7)}{3}=s+\frac{4}{3}). Tip: take the average even for symbolic zeroes.
If (p(x)=x^2-14x+53), why will the graph not cut the (x)-axis?
Correct answer: A
((x-7)^2+4) is always positive, so (p(x)=0) will not occur. Tip: adding a positive number to a square gives no real intersection.
If (p(x)=x(x+6)^4(x-3)^5), what will be the graph behavior at (x=3)?
Correct answer: A
((x-3)^5) is an odd-power factor, so the graph crosses at (x=3). Tip: at an odd power the graph usually crosses the axis.
If the zeroes of a parabola are (q-11) and (q+7), what will be its axis of symmetry?
Correct answer: A
The axis of symmetry is at the average of the zeroes, (\frac{(q-11)+(q+7)}{2}=q-2). Tip: take the midpoint even with symbols.
If the zeroes of a graph are (-12), (5), (14), which point does not show a zero on the graph?
Correct answer: C
((0,14)) has (y=14), so it is not on the (x)-axis. Tip: a zero point must have second coordinate (0).
If (p(x)=(x+9)(x-5)(x-12)), what will be the sign of (p(x)) for (5<x<12)?
Correct answer: B
In this interval the first two factors are positive and the third is negative, so the product is negative. Tip: check the sign of each factor separately.
If (p(x)=x^2-13x+k) has one zero (6), what will be the other zero and the (x)-axis intersections?
Correct answer: A
In the quadratic, the sum of zeroes is (13), so the other zero is (7). Tip: convert a zero into ((x,0)).
If a graph cuts the (x)-axis at (x=-12) and (x=12), what are the product and sum of the zeroes?
Correct answer: A
The zeroes are (-12) and (12), so the product is (-144) and the sum is (0). Tip: opposite zeroes have sum (0).
If \(p(x)=36x^2-49\), what are the x-axis intersections of its graph?
Correct answer: A
To find x-axis intersections, set \(p(x)=0\). Since \(36x^2-49=(6x-7)(6x+7)\), the equation \((6x-7)(6x+7)=0\) gives \(x=\frac{7}{6}\) or \(x=-\frac{7}{6}\). Therefore, the intercepts are \(\left(\frac{7}{6},0\right)\) and \(\left(-\frac{7}{6},0\right)\). Option C incorrectly uses the reciprocal \(\frac{6}{7}\). Exam tip: factor a difference of squares \(a^2-b^2\) as \((a-b)(a+b)\) before solving.
If the points meeting the (x)-axis are written as ((-11,0)), ((-11,0)), ((4,0)), ((4,0)), how many distinct real zeroes are there?
Correct answer: B
Repeated points give the same (x)-values, so the distinct zeroes are (-11) and (4). Tip: count the same (x)-value once.
A parabola cuts the (x)-axis at two points and cuts the (y)-axis at ((0,25)). What is the number of real zeroes?
Correct answer: B
Real zeroes are counted from (x)-axis intersections, not from the (y)-axis intercept. Tip: ((0,25)) does not show a zero.
If (p(x)=x^4-1296), what are the real (x)-axis intersections?
Correct answer: A
(x^4-1296=(x^2-36)(x^2+36)), and the real zeroes are only (\pm6). Tip: (x^2+36) gives no real zero.
If a polynomial cuts the (x)-axis at eight distinct points, what is the minimum possible degree?
Correct answer: C
For eight distinct real zeroes, the degree must be at least (8). Tip: the number of distinct zeroes cannot exceed the degree.
If (p(x)=x^2+16x+80), what is the graphical meaning of (p(x)=0)?
Correct answer: C
(x^2+16x+80=(x+8)^2+16), so there is no real zero. Tip: an always positive form gives no intersection.
If (p(x)=x^3-12x^2+35x), what is the set of zeroes of the graph?
Correct answer: A
(x^3-12x^2+35x=x(x-5)(x-7)), so the zeroes are (0), (5), (7). Tip: first take (x) as the common factor.
QUIZ COMPLETE