Problem 1:
The graph of a cubic polynomial
y = a x^{ 3} + b x^{ 2} + c x + d
is shown below. Find the coefficients a, b, c and d.
.
Solution to Problem 1:

This polynomial has a zero of multiplicity 1 at x = 2 and a zero of multiplicity 2 at x = 1. Hence the polynomial may be written as
y = a (x + 2)(x  1)^{ 2}

This polynomial has a y intercept (0 , 1). Hence
1 = a (0 + 2)(0  1)^{ 2}

Solve for a to obtain
a = 1 / 2

The polynomial may now be written as follows
y = (1 / 2) (x + 2)(x  1)^{ 2}

Expand to obtain
y = (1 / 2) x^{ 3} (3 / 2) x + 1

We now identify the coefficients as follows
a = 1/2 , b = 0, c = 3/2 and d = 1
Problem 2:
The graph of the polynomial
y = a x^{ 4} + b x^{ 3} + c x^{ 2} + d x + e
is shown below. Find the coefficients a, b, c, d and e.
.
Solution to Problem 2:

This polynomial has a zero of multiplicity 2 at x =  2 and a zero of multiplicity 2 at x = 2. Hence it may be written as
y = a (x + 2)^{ 2} (x  2)^{ 2}

We now use the y intercept at (0 , 2) to write the equation
2 = a (0 + 2)^{ 2} (0  2)^{ 2}

Solve the above for a to obtain
a =  1 / 8

We now write the polynomial as follows
y = (1 / 8) (x + 2)^{ 2} (x  2)^{ 2}

Expand
y = (1 / 8) x ^{ 4} + x^{ 2}  2

We now identify the coefficients
a = 1 / 8, b = 0, c = 1, d = 0, e = 1
Problem 3:
The polynomial
f(x) = x^{ 6} + 4 x^{ 5} + x^{ 4}  12 x^{ 3}  11 x^{ 2} + 4 x + 4
has a zero of multiplicity 2 at x =  2. Find the other real zeros.
Solution to Problem 3:

If f has a zero of multiplicity 2, then it may be written as follows
f(x) = (x + 2)^{ 2} Q(x)

Where Q(x) is a polynomial of degree 4 and may be found by division
Q(x) = f(x) / (x + 2)^{ 2} = x ^{ 4} 3 x ^{ 2} + 1

Polynomial f may now be written as
f(x) = (x + 2)^{ 2} (x ^{ 4} 3 x ^{ 2} + 1)

The remaining zeros of polynomial f may be found by solving the equation
x ^{ 4} 3 x ^{ 2} + 1 = 0

It is an equation of the quadratic type with solutions
( sqrt(5) + 1 ) / 2 , ( sqrt(5)  1 ) / 2 , (  sqrt(5)  1 ) / 2 , (  sqrt(5) + 1 ) / 2
Problem 4:
The polynomial
f(x) = 3 x^{ 4} + 5 x^{ 3}  17 x^{ 2}  25 x + 10
has irrational zeros at + sqrt(5) and  sqrt(5). Find the other zeros.
Solution to Problem 4:

polynomial f may be written as
f(x) = (x + sqrt(5)) (x  sqrt(5)) Q(x) = (x^{ 2}  5) Q(x)

Q(x) may be found by division
Q(x) = f(x) / (x^{ 2}  5) = 3 x ^{ 2} + 5 x  2

Hence f(x) may be written as
f(x) = (x^{ 2}  5) (3 x ^{ 2} + 5 x  2 )

The remaining zeros may found by solving the equation
3 x ^{ 2} + 5 x  2 = 0

Solve the above equation to find the remaining zeros of f.
2 and 1 / 3
Problem 5:
A polynomial of degree 4 has a positive leading coefficient and simple zeros (i.e. zeros of multiplicity 1) at x = 2, x =  2, x = 1 and x = 1. Is the y intercept of the graph of this polynomial positive or negative?
Solution to Problem 5:

All polynomials with degree 4 and positive leading coefficient will have a graph that rises to the left and to the right. And since the polynomial has two negative zeros and two positive zeros, then the only possibility for the y intercept is to be positive.
Problem 6:
The graph of polynomial p is shown below.
.
a  Is the degree of p even or odd?
b  Is the leading coefficient negative or positive?
c  Can you find the degree from the graph of p? Explain.
Solution to Problem 6:

a  odd

b  negative

c  No. The degree of a polynomial depends on the real and complex zeros. The graph shows only the real zeros. Hence, not enough information is given to find the degree of the polynomial.
Problem 7:
Give 4 different reasons why the graph below cannot be the graph of the polynomial p give by.
p(x) = x^{ 4}  x^{ 2} + 1
.
Solution to Problem 7:

1  The given polynomial has degree 4 and positive leading coefficient and the graph should rise on the left and right sides.

2  p(0) = 1, graph shows negative value.

3  the equation x^{ 4}  x^{ 2} + 1 = 0 has no solution which suggests that the polynomial p(x) = x^{ 4}  x^{ 2} + 1 has no zeros. The graph shows x intercepts.
 4  The graph has a zero of multiplicity 2, a zero of multiplicity 3 and and a zero of multiplicity 1. So the degree of the graphed polynomial should be at least 6. The given polynomial has degree 4.
More math problems with detailed solutions in this site.

