Solved on Jan 23, 2024

Identify which statements about the polynomial function $y=(x-5)^{2}(x+3)$ are true: $x$ -intercepts, degree, domain, range.

STEP 1

Assumptions
1. The given polynomial function is $y = (x-5)^2(x+3)$ .
2. We need to determine the truthfulness of the given statements about the polynomial function.

STEP 2

Analyze statement I: The $x$ -intercepts are -3 and 5.
To find the $x$ -intercepts, we set $y = 0$ and solve for $x$ .

STEP 3

Set the polynomial equal to zero to find the $x$ -intercepts.
$0 = (x-5)^2(x+3)$

STEP 4

Solve the equation $(x-5)^2(x+3) = 0$ by finding the values of $x$ that make each factor equal to zero.
$(x-5)^2 = 0 \quad \text{or} \quad (x+3) = 0$

STEP 5

Solve for $x$ in each case.
For $(x-5)^2 = 0$ , we have $x = 5$ .
For $(x+3) = 0$ , we have $x = -3$ .

STEP 6

Conclude the truthfulness of statement I.
Since the $x$ -intercepts are indeed -3 and 5, statement I is true.

STEP 7

Analyze statement II: The function has a degree of 3.
To determine the degree of the polynomial, we look at the highest power of $x$ when the polynomial is expanded.

STEP 8

Expand the polynomial to find the degree.
$y = (x-5)^2(x+3) = (x^2 - 10x + 25)(x + 3)$

STEP 9

Multiply out the factors to see the highest power of $x$ .
$y = x^3 - 10x^2 + 25x + 3x^2 - 30x + 75$

STEP 10

Combine like terms to find the standard form of the polynomial.
$y = x^3 - 7x^2 - 5x + 75$

STEP 11

Determine the degree of the polynomial.
The highest power of $x$ is 3, so the degree of the polynomial is 3.

STEP 12

Conclude the truthfulness of statement II.
Since the degree of the polynomial is 3, statement II is true.

STEP 13

Analyze statement III: The domain is $-3 \leq x \leq 5, x \in \mathbb{R}$ .
The domain of a polynomial function is the set of all possible $x$ -values for which the function is defined.

STEP 14

Determine the domain of the polynomial function.
For any polynomial function, the domain is all real numbers, $x \in \mathbb{R}$ , since polynomials are defined for all $x$ -values.

STEP 15

Conclude the truthfulness of statement III.
Since the domain of the polynomial is all real numbers and not restricted to $-3 \leq x \leq 5$ , statement III is false.

STEP 16

Analyze statement IV: The range is $y \in \mathbb{R}$ .
The range of a function is the set of all possible $y$ -values that the function can take.

STEP 17

Determine the range of the polynomial function.
Polynomial functions of odd degree have ranges that include all real numbers, $y \in \mathbb{R}$ , because they extend to both positive and negative infinity.

STEP 18

Conclude the truthfulness of statement IV.
Since the polynomial function has an odd degree and can take on any $y$ -value, statement IV is true.

STEP 19

Combine the conclusions of the truthfulness of each statement.
From our analysis, statements I, II, and IV are true, while statement III is false.

STEP 20

Select the correct answer based on the true statements.
The statements that are true are I, II, and IV.
The correct answer is c. I, II, and IV.

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Identify which statements about the polynomial function y=(x−5)2(x+3)y=(x-5)^{2}(x+3)y=(x−5)2(x+3) are true: xxx-intercepts, degree, domain, range.

STEP 1

Assumptions1. The given polynomial function is y=(x−5)2(x+3) y = (x-5)^2(x+3) y=(x−5)2(x+3).2. We need to determine the truthfulness of the given statements about the polynomial function.

STEP 2

Analyze statement I: The x x x-intercepts are -3 and 5.To find the x x x-intercepts, we set y=0 y = 0 y=0 and solve for x x x.

STEP 3

Set the polynomial equal to zero to find the x x x-intercepts.0=(x−5)2(x+3) 0 = (x-5)^2(x+3) 0=(x−5)2(x+3)

STEP 4

Solve the equation (x−5)2(x+3)=0 (x-5)^2(x+3) = 0 (x−5)2(x+3)=0 by finding the values of x x x that make each factor equal to zero.(x−5)2=0or(x+3)=0 (x-5)^2 = 0 \quad \text{or} \quad (x+3) = 0 (x−5)2=0or(x+3)=0

STEP 5

Solve for x x x in each case.For (x−5)2=0 (x-5)^2 = 0 (x−5)2=0, we have x=5 x = 5 x=5.For (x+3)=0 (x+3) = 0 (x+3)=0, we have x=−3 x = -3 x=−3.

STEP 6

Conclude the truthfulness of statement I.Since the x x x-intercepts are indeed -3 and 5, statement I is true.

STEP 7

Analyze statement II: The function has a degree of 3.To determine the degree of the polynomial, we look at the highest power of x x x when the polynomial is expanded.

STEP 8

Expand the polynomial to find the degree.y=(x−5)2(x+3)=(x2−10x+25)(x+3) y = (x-5)^2(x+3) = (x^2 - 10x + 25)(x + 3) y=(x−5)2(x+3)=(x2−10x+25)(x+3)

STEP 9

Multiply out the factors to see the highest power of x x x.y=x3−10x2+25x+3x2−30x+75 y = x^3 - 10x^2 + 25x + 3x^2 - 30x + 75 y=x3−10x2+25x+3x2−30x+75

STEP 10

Combine like terms to find the standard form of the polynomial.y=x3−7x2−5x+75 y = x^3 - 7x^2 - 5x + 75 y=x3−7x2−5x+75

STEP 11

Determine the degree of the polynomial.The highest power of x x x is 3, so the degree of the polynomial is 3.

STEP 12

Conclude the truthfulness of statement II.Since the degree of the polynomial is 3, statement II is true.

STEP 13

Analyze statement III: The domain is −3≤x≤5,x∈R -3 \leq x \leq 5, x \in \mathbb{R} −3≤x≤5,x∈R.The domain of a polynomial function is the set of all possible x x x-values for which the function is defined.

STEP 14

Determine the domain of the polynomial function.For any polynomial function, the domain is all real numbers, x∈R x \in \mathbb{R} x∈R, since polynomials are defined for all x x x-values.

STEP 15

Conclude the truthfulness of statement III.Since the domain of the polynomial is all real numbers and not restricted to −3≤x≤5 -3 \leq x \leq 5 −3≤x≤5, statement III is false.

STEP 16

Analyze statement IV: The range is y∈R y \in \mathbb{R} y∈R.The range of a function is the set of all possible y y y-values that the function can take.

STEP 17

Determine the range of the polynomial function.Polynomial functions of odd degree have ranges that include all real numbers, y∈R y \in \mathbb{R} y∈R, because they extend to both positive and negative infinity.

STEP 18

Conclude the truthfulness of statement IV.Since the polynomial function has an odd degree and can take on any y y y-value, statement IV is true.

STEP 19

Combine the conclusions of the truthfulness of each statement.From our analysis, statements I, II, and IV are true, while statement III is false.

STEP 20

Select the correct answer based on the true statements.The statements that are true are I, II, and IV.The correct answer is c. I, II, and IV.

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Identify which statements about the polynomial function $y=(x-5)^{2}(x+3)$ are true: $x$ -intercepts, degree, domain, range.

Assumptions
1. The given polynomial function is $y = (x-5)^2(x+3)$ .
2. We need to determine the truthfulness of the given statements about the polynomial function.

Analyze statement I: The $x$ -intercepts are -3 and 5.
To find the $x$ -intercepts, we set $y = 0$ and solve for $x$ .

Set the polynomial equal to zero to find the $x$ -intercepts.
$0 = (x-5)^2(x+3)$

Solve the equation $(x-5)^2(x+3) = 0$ by finding the values of $x$ that make each factor equal to zero.
$(x-5)^2 = 0 \quad \text{or} \quad (x+3) = 0$

Solve for $x$ in each case.
For $(x-5)^2 = 0$ , we have $x = 5$ .
For $(x+3) = 0$ , we have $x = -3$ .

Conclude the truthfulness of statement I.
Since the $x$ -intercepts are indeed -3 and 5, statement I is true.

Analyze statement II: The function has a degree of 3.
To determine the degree of the polynomial, we look at the highest power of $x$ when the polynomial is expanded.

Expand the polynomial to find the degree.
$y = (x-5)^2(x+3) = (x^2 - 10x + 25)(x + 3)$

Multiply out the factors to see the highest power of $x$ .
$y = x^3 - 10x^2 + 25x + 3x^2 - 30x + 75$

Combine like terms to find the standard form of the polynomial.
$y = x^3 - 7x^2 - 5x + 75$

Determine the degree of the polynomial.
The highest power of $x$ is 3, so the degree of the polynomial is 3.

Conclude the truthfulness of statement II.
Since the degree of the polynomial is 3, statement II is true.

Analyze statement III: The domain is $-3 \leq x \leq 5, x \in \mathbb{R}$ .
The domain of a polynomial function is the set of all possible $x$ -values for which the function is defined.

Determine the domain of the polynomial function.
For any polynomial function, the domain is all real numbers, $x \in \mathbb{R}$ , since polynomials are defined for all $x$ -values.

Conclude the truthfulness of statement III.
Since the domain of the polynomial is all real numbers and not restricted to $-3 \leq x \leq 5$ , statement III is false.

Analyze statement IV: The range is $y \in \mathbb{R}$ .
The range of a function is the set of all possible $y$ -values that the function can take.

Determine the range of the polynomial function.
Polynomial functions of odd degree have ranges that include all real numbers, $y \in \mathbb{R}$ , because they extend to both positive and negative infinity.

Conclude the truthfulness of statement IV.
Since the polynomial function has an odd degree and can take on any $y$ -value, statement IV is true.

Combine the conclusions of the truthfulness of each statement.
From our analysis, statements I, II, and IV are true, while statement III is false.

Select the correct answer based on the true statements.
The statements that are true are I, II, and IV.
The correct answer is c. I, II, and IV.