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) 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

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

(i) *x*^{2} – 2*x* – 8

(ii) 4*s*^{2} – 4*s* + 1

(iii) 6*x*^{2} – 3 – 7*x*

(iv) 4*u*^{2} + 8*u*

(v) *t*^{2} – 15

(vi) 3*x*^{2} –* x* – 4

(i) *x*^{2} – 2*x* – 8

= (*x* - 4) (*x* + 2)

The value of *x*^{2} – 2*x* – 8 is zero when *x* - 4 = 0 or *x* + 2 = 0, i.e., when *x* = 4 or *x* = -2

Therefore, the zeroes of *x*^{2} – 2*x* – 8 are 4 and -2.

Sum of zeroes = 4 + (-2) = 2 = -(-2)/1 = -(Coefficient of *x*)/Coefficient of *x*^{2}

Product of zeroes = 4 × (-2) = -8 = -8/1 = Constant term/Coefficient of *x*^{2}

(ii) 4*s*^{2} – 4*s* + 1

= (2*s*-1)^{2}

The value of 4*s*^{2} - 4*s* + 1 is zero when 2*s* - 1 = 0, i.e., s = 1/2

Therefore, the zeroes of 4s^{2} - 4s + 1 are 1/2 and 1/2.

Sum of zeroes = 1/2 + 1/2 = 1 = -(-4)/4 = -(Coefficient of *s)*/Coefficient of *s*^{2}

Product of zeroes = 1/2 × 1/2 = 1/4 = Constant term/Coefficient of *s*^{2}.

(iii) 6*x*^{2} – 3 – 7*x*

*= *6*x*^{2 }– 7*x *– 3

= (3*x* + 1) (2*x* - 3)

The value of 6*x*^{2} - 3 - 7*x* is zero when 3*x* + 1 = 0 or 2*x* - 3 = 0, i.e., *x* = -1/3 or *x* = 3/2

Therefore, the zeroes of 6*x*^{2} - 3 - 7*x* are -1/3 and 3/2.

Sum of zeroes = -1/3 + 3/2 = 7/6 = -(-7)/6 = -(Coefficient of *x*)/Coefficient of *x*^{2}

Product of zeroes = -1/3 × 3/2 = -1/2 = -3/6 = Constant term/Coefficient of *x*^{2}.

(iv) 4*u*^{2} + 8*u*

*= *4*u*^{2} + 8*u + *0

= 4*u*(*u* + 2)

The value of 4*u*^{2} + 8*u* is zero when 4*u* = 0 or *u* + 2 = 0, i.e., *u* = 0 or *u* = - 2

Therefore, the zeroes of 4*u*^{2} + 8*u* are 0 and - 2.

Sum of zeroes = 0 + (-2) = -2 = -(8)/4 = -(Coefficient of *u*)/Coefficient of *u*^{2}

Product of zeroes = 0 × (-2) = 0 = 0/4 = Constant term/Coefficient of *u*^{2}.

(v) *t*^{2} – 15

= *t*^{2 }- 0.*t* - 15

= (*t *- √15) (*t* + √15)

The value of *t*^{2} - 15 is zero when *t* - √15 = 0 or *t* + √15 = 0, i.e., when *t* = √15 or *t *= -√15

Therefore, the zeroes of *t*^{2} - 15 are √15 and -√15.Sum of zeroes = √15 + -√15 = 0 = -0/1 = -(Coefficient of *t*)/Coefficient of *t*^{2}

Product of zeroes = (√15) (-√15) = -15 = -15/1 = Constant term/Coefficient of *u*^{2}.

(vi) 3*x*^{2} –* x* – 4

= (3*x* - 4) (*x* + 1)

The value of 3*x*^{2} –* x* – 4 is zero when 3*x* - 4 = 0 and *x* + 1 = 0,i.e., when *x* = 4/3 or *x* = -1

Therefore, the zeroes of 3*x*^{2} –* x* – 4 are 4/3 and -1.

Sum of zeroes = 4/3 + (-1) = 1/3 = -(-1)/3 = -(Coefficient of *x*)/Coefficient of *x*^{2}

Product of zeroes = 4/3 × (-1) = -4/3 = Constant term/Coefficient of *x*^{2}.

Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.

(i) 1/4 , -1

(ii) √2 , 1/3

(iii) 0, √5

(iv) 1,1

(v) -1/4 ,1/4

(vi) 4,1

(i) 1/4 , -1

Let the polynomial be *ax*^{2} + *bx* + *c*, and its zeroes be α and ß

α + ß = 1/4 = -*b*/*a*

αß = -1 = -4/4 = *c*/*a*

If *a* = 4, then *b* = -1, *c* = -4

Therefore, the quadratic polynomial is 4*x*^{2} - *x* -4.

(ii) √2 , 1/3

Let the polynomial be *ax*^{2} + *bx* + *c*, and its zeroes be α and ß

α + ß = √2 = 3√2/3 = -*b*/*a*

αß = 1/3 = *c*/*a*

If *a* = 3, then *b* = -3√2, *c* = 1

Therefore, the quadratic polynomial is 3*x*^{2} -3√2*x* +1.

(iii) 0, √5

Let the polynomial be *ax*^{2} + *bx* + *c*, and its zeroes be α and ß

α + ß = 0 = 0/1 = -*b*/*a*

αß = √5 = √5/1 = *c*/*a*

If *a* = 1, then *b* = 0, *c* = √5

Therefore, the quadratic polynomial is *x*^{2} + √5.

(iv) 1, 1

Let the polynomial be *ax*^{2} + *bx* + *c*, and its zeroes be α and ß

α + ß = 1 = 1/1 = -*b*/*a*

αß = 1 = 1/1 = *c*/*a*

If *a* = 1, then *b* = -1, *c* = 1

Therefore, the quadratic polynomial is *x*^{2} - *x* +1.

(v) -1/4 ,1/4

Let the polynomial be *ax*^{2} + *bx* + *c*, and its zeroes be α and ß

α + ß = -1/4 = -*b*/*a*

αß = 1/4 = *c*/*a*

If *a* = 4, then *b* = 1, *c* = 1

Therefore, the quadratic polynomial is 4*x*^{2} + *x* +1.

(vi) 4,1

Let the polynomial be *ax*^{2} + *bx* + *c*, and its zeroes be α and ß

α + ß = 4 = 4/1 = -*b*/*a*

αß = 1 = 1/1 = *c*/*a*

If *a* = 1, then *b* = -4, *c* = 1

Therefore, the quadratic polynomial is *x*^{2} - 4*x* +1.

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