Question

precalculus unit 3 test 2025
name
date
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multiple choice: circle the letter of the correct answer and write the letter next to the number.

1. use the remainder theorem to find which of the following is not a factor of the polynomial

x³ + 3x² - 18x - 40
a) x + 5
b) x + 2
c) x - 4
d) x - 5

1. find the discriminant of 4x² - 3x - 5 = 0.

a) -89
b) -√89
c) 89
d) √89

1. determine which of the following equations has roots -1, -5, and 4.

a) x³ + 29x - 20 = 0
b) x³ - 2x² + 19x + 20 = 0
c) x³ - 8x² - 11x - 20 = 0
d) x³ + 2x² - 19x - 20 = 0

1. list all possible rational zeros of the function:

f(x) = -6x⁴ + 81x³ + x + 5
a) ±1, ±½, ±⅓, ±⅙, ±5, ±⁵⁄₂, ±⁵⁄₃, ±⁵⁄₆
b) ±1, ±⅙, ±2, ±⅔, ±3, ±³⁄₅, ±6, ±⁶⁄₅
c) ±1, ±5, ±6, ±⅕, ±⅙
d) ±5, ±⁵⁄₆, ±6, ±⁶⁄₅

1. find the vertex of the following quadratic:

y = 2x² - 12x + 11
a) (3, -7)
b) (-3, 65)
c) (3, 65)
d) (0, 11)
for numbers 6 - 11, show your work & circle final answers.

1. solve for all real roots by fully factoring the polynomial:

g(x) = x³ - 9x² + 20x
fully factored form:
roots:

1. write the polynomial of least degree with roots of 5, ±√3
2. describe the end behavior of the function and sketch a potential graph.

f(x) = -2x³ + 6x² - 5x + 4
as x → -∞, f(x) →
as x → ∞, f(x) →

Explanation:

Question 6: Solve for all real roots by FULLY FACTORING the polynomial \( g(x) = x^3 - 9x^2 + 20x \)

Step 1: Factor out the greatest common factor (GCF)

The GCF of \( x^3 \), \( -9x^2 \), and \( 20x \) is \( x \). So we factor out \( x \):
\( g(x) = x(x^2 - 9x + 20) \)

Step 2: Factor the quadratic trinomial

We need to find two numbers that multiply to \( 20 \) and add up to \( -9 \). Those numbers are \( -4 \) and \( -5 \). So we factor \( x^2 - 9x + 20 \) as \( (x - 4)(x - 5) \):
\( g(x) = x(x - 4)(x - 5) \)

Step 3: Find the roots

To find the roots, we set \( g(x) = 0 \):
\( x(x - 4)(x - 5) = 0 \)
Using the zero - product property (if \( ab = 0 \), then either \( a = 0 \), \( b = 0 \), or both), we get:
\( x = 0 \) or \( x - 4 = 0 \) (which gives \( x = 4 \)) or \( x - 5 = 0 \) (which gives \( x = 5 \))

Fully factored form:

\( g(x)=x(x - 4)(x - 5) \)

Roots:

\( x = 0 \), \( x = 4 \), \( x = 5 \)

Question 7: Write the polynomial of least degree with roots of \( 5 \), \( \pm\sqrt{3} \)

Step 1: Recall the factor theorem

If \( r \) is a root of a polynomial, then \( (x - r) \) is a factor of the polynomial.

Step 2: Find the factors corresponding to each root

• For the root \( x = 5 \), the factor is \( (x - 5) \).
• For the root \( x=\sqrt{3} \), the factor is \( (x-\sqrt{3}) \).
• For the root \( x = -\sqrt{3} \), the factor is \( (x+\sqrt{3}) \).

Step 3: Multiply the factors

First, multiply \( (x - \sqrt{3})(x+\sqrt{3}) \). Using the difference of squares formula \( (a - b)(a + b)=a^2 - b^2 \), we have \( (x - \sqrt{3})(x+\sqrt{3})=x^2-(\sqrt{3})^2=x^2 - 3 \).

Then multiply this result by \( (x - 5) \):
\( (x - 5)(x^2 - 3)=x(x^2 - 3)-5(x^2 - 3)=x^3-3x - 5x^2 + 15=x^3-5x^2-3x + 15 \)

Step 1: Recall the rule for end - behavior of polynomials

For a polynomial function of the form \( f(x)=a_nx^n+a_{n - 1}x^{n - 1}+\cdots+a_1x + a_0 \), the end - behavior is determined by the leading term \( a_nx^n \), where \( n \) is the degree of the polynomial and \( a_n \) is the leading coefficient.

Step 2: Identify the leading term of \( f(x) \)

For the function \( f(x)=-2x^3+6x^2 - 5x + 4 \), the leading term is \( -2x^3 \). Here, the degree \( n = 3 \) (which is odd) and the leading coefficient \( a_n=-2 \) (which is negative).

Step 3: Determine the end - behavior based on the leading term

• When the degree \( n \) is odd and the leading coefficient \( a_n<0 \):
• As \( x

ightarrow-\infty \), we consider the behavior of \( -2x^3 \). If \( x
ightarrow-\infty \), then \( x^3
ightarrow-\infty \), and \( -2x^3=-2\times(-\infty)=\infty \). So as \( x
ightarrow-\infty \), \( f(x)
ightarrow\infty \).

• As \( x

ightarrow\infty \), \( x^3
ightarrow\infty \), and \( -2x^3=-2\times\infty = -\infty \). So as \( x
ightarrow\infty \), \( f(x)
ightarrow-\infty \).

Answer:

The polynomial of least degree is \( x^3-5x^2 - 3x + 15 \)

Question 8: Describe the end behavior of the function \( f(x)=-2x^3+6x^2 - 5x + 4 \) and find \( \text{As }x

ightarrow-\infty,f(x)
ightarrow\text{?} \) and \( \text{As }x
ightarrow\infty,f(x)
ightarrow\text{?} \)