Chapter P.5, Problem 58E

Solution

a.

The quadratic polynomial with two real zeros.

  $x^2+4x+3$

Given:

The discriminant of a quadratic polynomial $a{x}^2+bx+c$ is $b^2-4ac$ .

Concept used:

• The discriminant is given by, $D=b^2-4ac$
• For a quadratic equation $a{x}^2+bx+c=0$ the number of roots depend on:
• If discriminant is negative, $b^2-4ac<0$, then there is no real roots
• If discriminant is zero, $b^2-4ac=0$, then there are two same real roots
• If discriminant is positive, $b^2-4ac>0$, then there are two distinct real roots

Calculation:

Recall that a quadratic polynomial has two real zeros when the discriminant is positive, that is, $b^2-4ac>0$ .

So, in order to find the quadratic polynomial, find such a , b , and c for which the discriminant is positive. One such set of values is $a=1,b=4,c=3$ .

Thus, one such quadratic polynomial with two real zeros is $x^2+4x+3$ .

The graph of the polynomial $y=x^2+4x+3$ is shown below:

  

b.

The quadratic polynomial with exactly one real zero.

  $x^2+4x+4$

Given:

The discriminant of a quadratic polynomial $a{x}^2+bx+c$ is $b^2-4ac$ .

Concept used:

• The discriminant is given by, $D=b^2-4ac$
• For a quadratic equation $a{x}^2+bx+c=0$ the number of roots depend on:
• If discriminant is negative, $b^2-4ac<0$, then there is no real roots
• If discriminant is zero, $b^2-4ac=0$, then there are two same real roots
• If discriminant is positive, $b^2-4ac>0$, then there are two distinct real roots

Calculation:

Recall that a quadratic polynomial has exactly one real zero when the discriminant is 0, that is, $b^2-4ac=0$ .

So, in order to find the quadratic polynomial, find such a , b , and c for which the discriminant is 0. One such set of values is $a=1,b=4,c=4$ .

Thus, one such quadratic polynomial with two real zeros is $x^2+4x+4$ .

The graph of the polynomial $y=x^2+4x+4$ is shown below:

  

c.

The quadratic polynomial with no real zeros.

  $x^2+4x+5$ .

Given:

The discriminant of a quadratic polynomial $a{x}^2+bx+c$ is $b^2-4ac$ .

Concept used:

• The discriminant is given by, $D=b^2-4ac$
• For a quadratic equation $a{x}^2+bx+c=0$ the number of roots depend on:
• If discriminant is negative, $b^2-4ac<0$, then there is no real roots
• If discriminant is zero, $b^2-4ac=0$, then there are two same real roots
• If discriminant is positive, $b^2-4ac>0$, then there are two distinct real roots

Calculation:

Recall that a quadratic polynomial has no real zeros when the discriminant is negative, that is, $b^2-4ac<0$ .

So, in order to find the quadratic polynomial, find such a , b , and c for which the discriminant is negative. One such set of values is $a=1,b=4,c=5$ .

Thus, one such quadratic polynomial with two real zeros is $x^2+4x+5$ .

The graph of the polynomial $y=x^2+4x+5$ is shown below: