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.
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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.
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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.
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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
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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.
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