Question

find the real solutions by factoring
x³ - 49x = 0
what is the solution set? select the correct choice below and fill in any answer boxes within your choice.
a. { } (use a comma to separate answers as needed. type an integer or a simplified fraction.)
b. there are no solutions.

1. form a polynomial whose real zeros and degree are given.

zeros: - 3, 0, 5, degree: 3
type a polynomial with integer coefficients and a leading coefficient of 1.
f(x) = (simplify your answer.)

1. for the polynomial function below,

f(x) = - 2(x - 1)(x + 5)²
a) find any real zeros of f. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the real zero(s) of f is/are (type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.)
b. there are no real zeros.
the multiplicity of the larger zero is (type a whole number.)
the multiplicity of the smaller zero is (type a whole number.)
b) the graph (1) the x - axis at the larger x - intercept.
the graph (2) the x - axis at the smaller x - intercept.
c) the maximum number of turning points on the graph is (type a whole number.)
d) type the power function that the graph of f resembles for large values of |x|.
y =

Explanation:

Problem 17: Form a polynomial with given zeros and degree

Step 1: Recall the polynomial form from zeros

If the zeros of a polynomial are \( r_1, r_2, r_3, \dots \), then the polynomial can be written in factored form as \( f(x) = a(x - r_1)(x - r_2)(x - r_3)\dots \), where \( a \) is the leading coefficient. Here, the zeros are \( -3, 0, 5 \) and the degree is 3, and the leading coefficient \( a = 1 \).
So, the factored form is \( f(x)=(x - (-3))(x - 0)(x - 5)=x(x + 3)(x - 5) \)

Step 2: Expand the factored form

First, multiply \( (x + 3)(x - 5) \):
\( (x + 3)(x - 5)=x^2-5x + 3x-15=x^2-2x - 15 \)
Then multiply by \( x \):
\( f(x)=x(x^2-2x - 15)=x^3-2x^2-15x \)

Answer:

\( f(x)=x^3 - 2x^2 - 15x \)

Problem 18: Analyze the polynomial \( f(x)=-2(2x - 1)(x + 5)^2 \)

(a) Find real zeros

Step 1: Set \( f(x) = 0 \)

\( -2(2x - 1)(x + 5)^2=0 \)
Since a product is zero when at least one factor is zero, we have:
\( 2x - 1 = 0 \) or \( (x + 5)^2=0 \)

Step 2: Solve for \( x \)

For \( 2x - 1 = 0 \), we get \( 2x=1\implies x=\frac{1}{2} \)
For \( (x + 5)^2=0 \), we get \( x=-5 \) (with multiplicity 2)
So the real zeros are \( x=\frac{1}{2}, x = - 5 \)

(b) Multiplicity and graph behavior at intercepts

• For the larger zero \( x=\frac{1}{2} \): The factor is \( (2x - 1) \) which has an exponent of 1, so multiplicity is 1. When multiplicity is odd, the graph crosses the x - axis.
• For the smaller zero \( x=-5 \): The factor is \( (x + 5)^2 \) which has an exponent of 2, so multiplicity is 2. When multiplicity is even, the graph touches the x - axis (bounces off).

(c) Maximum number of turning points

The degree of the polynomial \( f(x)=-2(2x - 1)(x + 5)^2=-2(2x - 1)(x^2 + 10x + 25)=-2(2x^3+20x^2 + 50x-x^2-10x - 25)=-2(2x^3 + 19x^2+40x - 25)=-4x^3-38x^2 - 80x + 50 \)
The degree \( n = 3 \). The maximum number of turning points of a polynomial of degree \( n \) is \( n - 1 \). So for \( n = 3 \), the maximum number of turning points is \( 3-1 = 2 \)

(d) End - behavior power function

The leading term of the polynomial \( f(x) \) is \( -4x^3 \). For large \( |x| \), the graph of a polynomial resembles the graph of its leading term. The leading term \( -4x^3 \) resembles the power function \( y=-x^3 \) (we can ignore the coefficient - 4 for the end - behavior shape, just consider the leading term's degree and sign. The degree is 3 and the leading coefficient is negative, so the power function is \( y=-x^3 \))

Final Answers:

1. \( \boldsymbol{f(x)=x^3 - 2x^2 - 15x} \)

1. (a) Real zeros: \( \boldsymbol{x=\frac{1}{2}, x=-5} \)

(b) Multiplicity of larger zero (\( x = \frac{1}{2} \)): 1; Multiplicity of smaller zero (\( x=-5 \)): 2; Graph (1) (at \( x=\frac{1}{2} \)) crosses the x - axis; Graph (2) (at \( x=-5 \)) touches the x - axis.
(c) Maximum number of turning points: \( \boldsymbol{2} \)
(d) Power function: \( \boldsymbol{y=-x^3} \)