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Standard X

Mathematics

Polynomial

The graph of ...

Question

The graph of the polynomial p(x) intersects the x-axis three times in distinct points, then which of the following could be an expression for p(x):

4 - 4x - x² + x³

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3x + 3

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3x² + 3x - 3

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x² - 9

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Solution

The correct option is A
4 - 4x - x² + x³

Since, the graph of the polynomial p(x) intersects the x-axis at three distinct points, it will have 3 distinct roots. Hence, it will be a cubic polynomial.

Thus, amongst the given alternatives the polynomial p(x) can be 4 - 4x - x² + x³.

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Similar questions

Now in each of the following, find the quotient and remainder on dividing p(x) by q(x) and write then in the form p(x) = u(x) q(x) + v(x).

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

Q. Classify the following polynomials based on their degree.
(i)

p (x) = 3

(ii)

p (y) = \frac{5}{2} y^{2} + 1

(iii)

p (x) = 2 x^{3} - x^{2} + 4 x + 1

(iv)

p (x) = 3 x^{2}

(v)

p (x) = x + 3

(vi)

p (x) = - 7

(vii)

p (x) = x^{3} + 1

(viii)

p (x) = 5 x^{2} - 3 x + 2

(ix)

p (x) = 4 x

(x)

p (x) = \frac{3}{2}

(xi)

p (x) = \sqrt{3} x + 1

(xii)

p (y) = y^{3} + 3 y

Identify polynomials in the following :

(i) f(x) = $4 x^{3} - x^{2}$ - 3x + 7

(ii) g(x) = $2 x^{3} - 3 x^{2} + \sqrt{x} - 1$

(iii) p(x) = $\frac{2}{7} x^{2} - \frac{7}{4} x + 9$

(iv) q(x) = $2 x^{2} - 3 x + \frac{4}{x} + 2$

(v) h(x) = $x^{4} - x^{\frac{3}{2}} + x - 1$

(vi) f(x) = 2 + $\frac{3}{x} + 4 x$

Q. Divide the polynomial

p (x)

by the polynomial

g (x)

and find the quotient and remainder in each of the following:
(i)

p (x) = x^{3} - 3 x^{2} + 5 x - 3, g (x) = x^{2} - 2

(ii)

p (x) = x^{4} - 3 x^{2} + 4 x + 5, g (x) = x^{2} + 1 - x

(iii)

p (x) = x^{4} - 5 x + 6, g (x) = 2 - x^{2}

Q. If 2 is a zero of the polynomial

p (x)

and one zero is half of the other zero. Which of the following expressions can

p (x)

be?

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The graph of the polynomial p(x) intersects the x-axis three times in distinct points, then which of the following could be an expression for p(x):

The correct option is A 4 - 4x - x2 + x3Since, the graph of the polynomial p(x) intersects the x-axis at three distinct points, it will have 3 distinct roots. Hence, it will be a cubic polynomial. Thus, amongst the given alternatives the polynomial p(x) can be 4 - 4x - x2 + x3.

The correct option is A
4 - 4x - x² + x³

Since, the graph of the polynomial p(x) intersects the x-axis at three distinct points, it will have 3 distinct roots. Hence, it will be a cubic polynomial.

Thus, amongst the given alternatives the polynomial p(x) can be 4 - 4x - x² + x³.