Polynomials Class 10

Polynomials Class 10 QUESTION : 1:

The graphs of y = p(x) are given in following figure, for some polynomials p(x). Find the number of zeroes of p(x), in each case.

(i)

(ii)

(iii)

(iv)

(v)

(v)

(i) The number of zeroes is 0 as the graph does not cut the x-axis at any point.

(ii) The number of zeroes is 1 as the graph intersects the x-axis at only 1 point.

(iii) The number of zeroes is 3 as the graph intersects the x-axis at 3 points.

(iv) The number of zeroes is 2 as the graph intersects the x-axis at 2 points.

(v) The number of zeroes is 4 as the graph intersects the x-axis at 4 points.

(vi) The number of zeroes is 3 as the graph intersects the x-axis at 3 points.

Polynomials Class 10  Polynomials Class 10 QUESTION : 1:

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.

The value of is zero when x − 4 = 0 or x + 2 = 0, i.e., when x = 4 or x = −2

Therefore, the zeroes of are 4 and −2.

Sum of zeroes =

Product of zeroes

The value of 4s² − 4s + 1 is zero when 2s − 1 = 0, i.e.,

Therefore, the zeroes of 4s² − 4s + 1 areand.

Sum of zeroes =

Product of zeroes

The value of 6x² − 3 − 7x is zero when 3x + 1 = 0 or 2x − 3 = 0, i.e., or

Therefore, the zeroes of 6x² − 3 − 7x are.

Sum of zeroes =

Product of zeroes =

The value of 4u² + 8u is zero when 4u = 0 or u + 2 = 0, i.e., u = 0 or u = −2

Therefore, the zeroes of 4u² + 8u are 0 and −2.

Sum of zeroes =

Product of zeroes =

The value of t² − 15 is zero when or , i.e., when

Therefore, the zeroes of t² − 15 are and.

Sum of zeroes =

Product of zeroes =

The value of 3x² − x − 4 is zero when 3x − 4 = 0 or x + 1 = 0, i.e., when or x = −1

Therefore, the zeroes of 3x² − x − 4 are and −1.

Sum of zeroes =

Product of zeroes

Polynomials Class 10 Polynomials Class 10 QUESTION : 1:

Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following:

(i)

(ii)

(iii)

Quotient = x − 3

Remainder = 7x − 9

Quotient = x² + x − 3

Remainder = 8

Quotient = −x² − 2

Remainder = −5x +10

Polynomials Class 10 QUESTION : 2:

Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial:

=

Since the remainder is 0,

Hence, is a factor of .

Since the remainder is 0,

Hence, is a factor of .

Since the remainder ,

Hence, is not a factor of .