Answer
Verified
430.2k+ views
Hint: A quadratic polynomial is a polynomial of degree 2. An equation involving a quadratic polynomial is a quadratic equation. The standard form is
\[a{{x}^{2}}+bx+c=0\]
with a, b, and c being constants or numerical coefficients, and x is an unknown variable. One absolute rule is that the first constant "a" cannot be a zero.
Complete step-by-step solution:
The x-intercepts are the points at which the parabola crosses the x-axis. If they exist, the x-intercepts represent the zeros, or roots, of the quadratic function.
Now, in the question, it is mentioned that a given quadratic polynomial has no zero.
Let us assume that the quadratic polynomial is the form
\[p{{x}^{2}}+qx+r=0\]
We also know that the roots of the above quadratic polynomial are
X= \[\dfrac{-q+\sqrt{{{q}^{2}}-4pr}}{2p},\dfrac{-q-\sqrt{{{q}^{2}}-4pr}}{2p}\]
We also know that for a quadratic equation if x-intercept exist it means that the given equation has roots or also it can be concluded that the graph of the polynomial crosses the x- axis.
But, from our question, it is said that the quadratic polynomial has no zero, which means there exists no x for which the graph intersects the x-axis.
In the diagram below we can see an example of a parabola which is the graph of a quadratic polynomial, it is given in the question that it doesn’t have a zero which implies that the quadratic equation had no roots and hence it doesn’t intersect the x-axis. As the roots of the polynomial lie on the x-axis.
In the diagram below the parabola touches the x-axis, from which we can conclude that the parabola would have two equal roots.
In option A it is given that the graph touches the x-axis at one point, which would be false because if the graph intersects the x-axis at one point then y=0 will be zero for some x.
Hence option C is correct which is the graph doesn’t intersect the x-axis at any point.
Note: Students should read the question properly or they might get confused with what is actually given. A quadratic polynomial has at most two real roots so the maximum no of times the graph will intersect the x-axis is two and the minimum would be zero when the equation has no roots.
\[a{{x}^{2}}+bx+c=0\]
with a, b, and c being constants or numerical coefficients, and x is an unknown variable. One absolute rule is that the first constant "a" cannot be a zero.
Complete step-by-step solution:
The x-intercepts are the points at which the parabola crosses the x-axis. If they exist, the x-intercepts represent the zeros, or roots, of the quadratic function.
Now, in the question, it is mentioned that a given quadratic polynomial has no zero.
Let us assume that the quadratic polynomial is the form
\[p{{x}^{2}}+qx+r=0\]
We also know that the roots of the above quadratic polynomial are
X= \[\dfrac{-q+\sqrt{{{q}^{2}}-4pr}}{2p},\dfrac{-q-\sqrt{{{q}^{2}}-4pr}}{2p}\]
We also know that for a quadratic equation if x-intercept exist it means that the given equation has roots or also it can be concluded that the graph of the polynomial crosses the x- axis.
But, from our question, it is said that the quadratic polynomial has no zero, which means there exists no x for which the graph intersects the x-axis.
In the diagram below we can see an example of a parabola which is the graph of a quadratic polynomial, it is given in the question that it doesn’t have a zero which implies that the quadratic equation had no roots and hence it doesn’t intersect the x-axis. As the roots of the polynomial lie on the x-axis.
In the diagram below the parabola touches the x-axis, from which we can conclude that the parabola would have two equal roots.
In option A it is given that the graph touches the x-axis at one point, which would be false because if the graph intersects the x-axis at one point then y=0 will be zero for some x.
Hence option C is correct which is the graph doesn’t intersect the x-axis at any point.
Note: Students should read the question properly or they might get confused with what is actually given. A quadratic polynomial has at most two real roots so the maximum no of times the graph will intersect the x-axis is two and the minimum would be zero when the equation has no roots.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Which are the Top 10 Largest Countries of the World?
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you graph the function fx 4x class 9 maths CBSE
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Difference Between Plant Cell and Animal Cell
Give 10 examples for herbs , shrubs , climbers , creepers
Write a stanza wise summary of money madness class 11 english CBSE
Change the following sentences into negative and interrogative class 10 english CBSE