Math / Algebra

QuestionQuestion 16
Use the rational zeros theorem to list all possible zeros of the function $f(x)=5 x^{3}-5 x^{2}-4 x+2$ . Enter the possible zeros separated by commas. You do not need to factor the polynomial.

Studdy Solution

STEP 1

What is this asking? We need to find all the possible rational zeros of the given polynomial using the rational zeros theorem. Watch out! The theorem only gives us possible rational zeros, not the actual zeros!
Don't forget the plus and minus signs for each possible zero.

STEP 2

1. Identify the leading coefficient and constant term.
2. List the factors of the constant term.
3. List the factors of the leading coefficient.
4. Form all possible rational zeros.

STEP 3

Our function is $f(x) = 5x^3 - 5x^2 - 4x + 2$ .
The leading coefficient is the number multiplying the highest power of $x$ , which is 5.
The constant term is the number without any $x$ , which is 2.

STEP 4

The factors of 2 are the numbers that divide 2 evenly.
These are 1 and 2.
Remember, factors can be positive or negative, so we have $\pm 1$ and $\pm 2$ .

STEP 5

The factors of 5 are the numbers that divide 5 evenly.
These are 1 and 5.
Again, remember the plus/minus, giving us $\pm 1$ and $\pm 5$ .

STEP 6

The rational zeros theorem says that any possible rational zero is of the form $\pm \frac{\text{factor of constant term}}{\text{factor of leading coefficient}}$ .

STEP 7

Let's create all the combinations!
Dividing the factors of 2 ( $\pm 1$ and $\pm 2$ ) by $\pm 1$ gives us $\pm \frac{1}{1} = \pm 1$ and $\pm \frac{2}{1} = \pm 2$ .

STEP 8

Dividing the factors of 2 ( $\pm 1$ and $\pm 2$ ) by $\pm 5$ gives us $\pm \frac{1}{5}$ and $\pm \frac{2}{5}$ .

STEP 9

So, all possible rational zeros are $\pm 1$ , $\pm 2$ , $\pm \frac{1}{5}$ , and $\pm \frac{2}{5}$ .

STEP 10

The possible rational zeros of $f(x) = 5x^3 - 5x^2 - 4x + 2$ are $1, -1, 2, -2, \frac{1}{5}, -\frac{1}{5}, \frac{2}{5}, -\frac{2}{5}$ .

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QuestionQuestion 16
Use the rational zeros theorem to list all possible zeros of the function $f(x)=5 x^{3}-5 x^{2}-4 x+2$ . Enter the possible zeros separated by commas. You do not need to factor the polynomial.

Studdy Solution

STEP 1

What is this asking? We need to find all the possible rational zeros of the given polynomial using the rational zeros theorem. Watch out! The theorem only gives us possible rational zeros, not the actual zeros!
Don't forget the plus and minus signs for each possible zero.

STEP 2

1. Identify the leading coefficient and constant term.
2. List the factors of the constant term.
3. List the factors of the leading coefficient.
4. Form all possible rational zeros.

STEP 3

Our function is $f(x) = 5x^3 - 5x^2 - 4x + 2$ .
The leading coefficient is the number multiplying the highest power of $x$ , which is 5.
The constant term is the number without any $x$ , which is 2.

STEP 4

The factors of 2 are the numbers that divide 2 evenly.
These are 1 and 2.
Remember, factors can be positive or negative, so we have $\pm 1$ and $\pm 2$ .

STEP 5

The factors of 5 are the numbers that divide 5 evenly.
These are 1 and 5.
Again, remember the plus/minus, giving us $\pm 1$ and $\pm 5$ .

STEP 6

The rational zeros theorem says that any possible rational zero is of the form $\pm \frac{\text{factor of constant term}}{\text{factor of leading coefficient}}$ .

STEP 7

Let's create all the combinations!
Dividing the factors of 2 ( $\pm 1$ and $\pm 2$ ) by $\pm 1$ gives us $\pm \frac{1}{1} = \pm 1$ and $\pm \frac{2}{1} = \pm 2$ .

STEP 8

Dividing the factors of 2 ( $\pm 1$ and $\pm 2$ ) by $\pm 5$ gives us $\pm \frac{1}{5}$ and $\pm \frac{2}{5}$ .

STEP 9

So, all possible rational zeros are $\pm 1$ , $\pm 2$ , $\pm \frac{1}{5}$ , and $\pm \frac{2}{5}$ .

STEP 10

The possible rational zeros of $f(x) = 5x^3 - 5x^2 - 4x + 2$ are $1, -1, 2, -2, \frac{1}{5}, -\frac{1}{5}, \frac{2}{5}, -\frac{2}{5}$ .

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QuestionQuestion 16Use the rational zeros theorem to list all possible zeros of the function f(x)=5x3−5x2−4x+2f(x)=5 x^{3}-5 x^{2}-4 x+2f(x)=5x3−5x2−4x+2. Enter the possible zeros separated by commas. You do not need to factor the polynomial.

Studdy Solution

STEP 1

What is this asking? We need to find all the *possible* rational zeros of the given polynomial using the rational zeros theorem. Watch out! The theorem only gives us *possible* rational zeros, not the *actual* zeros!Don't forget the plus and minus signs for each possible zero.

STEP 2

1. Identify the leading coefficient and constant term.2. List the factors of the constant term.3. List the factors of the leading coefficient.4. Form all possible rational zeros.

STEP 3

Our function is f(x)=5x3−5x2−4x+2f(x) = 5x^3 - 5x^2 - 4x + 2f(x)=5x3−5x2−4x+2.The **leading coefficient** is the number multiplying the highest power of xxx, which is **5**.The **constant term** is the number without any xxx, which is **2**.

STEP 4

The factors of **2** are the numbers that divide **2** evenly.These are **1** and **2**.Remember, factors can be positive or negative, so we have ±1\pm 1±1 and ±2\pm 2±2.

STEP 5

The factors of **5** are the numbers that divide **5** evenly.These are **1** and **5**.Again, remember the plus/minus, giving us ±1\pm 1±1 and ±5\pm 5±5.

STEP 6

The rational zeros theorem says that any *possible* rational zero is of the form ±factor of constant termfactor of leading coefficient\pm \frac{\text{factor of constant term}}{\text{factor of leading coefficient}}±factor of leading coefficientfactor of constant term​.

STEP 7

Let's create all the combinations!Dividing the factors of **2** (±1\pm 1±1 and ±2\pm 2±2) by ±1\pm 1±1 gives us ±11=±1\pm \frac{1}{1} = \pm 1±11​=±1 and ±21=±2\pm \frac{2}{1} = \pm 2±12​=±2.

STEP 8

Dividing the factors of **2** (±1\pm 1±1 and ±2\pm 2±2) by ±5\pm 5±5 gives us ±15\pm \frac{1}{5}±51​ and ±25\pm \frac{2}{5}±52​.

STEP 9

So, all possible rational zeros are ±1\pm 1±1, ±2\pm 2±2, ±15\pm \frac{1}{5}±51​, and ±25\pm \frac{2}{5}±52​.

STEP 10

The possible rational zeros of f(x)=5x3−5x2−4x+2f(x) = 5x^3 - 5x^2 - 4x + 2f(x)=5x3−5x2−4x+2 are 1,−1,2,−2,15,−15,25,−251, -1, 2, -2, \frac{1}{5}, -\frac{1}{5}, \frac{2}{5}, -\frac{2}{5}1,−1,2,−2,51​,−51​,52​,−52​.

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QuestionQuestion 16
Use the rational zeros theorem to list all possible zeros of the function $f(x)=5 x^{3}-5 x^{2}-4 x+2$ . Enter the possible zeros separated by commas. You do not need to factor the polynomial.

What is this asking? We need to find all the possible rational zeros of the given polynomial using the rational zeros theorem. Watch out! The theorem only gives us possible rational zeros, not the actual zeros!
Don't forget the plus and minus signs for each possible zero.

1. Identify the leading coefficient and constant term.
2. List the factors of the constant term.
3. List the factors of the leading coefficient.
4. Form all possible rational zeros.

Our function is $f(x) = 5x^3 - 5x^2 - 4x + 2$ .
The leading coefficient is the number multiplying the highest power of $x$ , which is 5.
The constant term is the number without any $x$ , which is 2.

The factors of 2 are the numbers that divide 2 evenly.
These are 1 and 2.
Remember, factors can be positive or negative, so we have $\pm 1$ and $\pm 2$ .

The factors of 5 are the numbers that divide 5 evenly.
These are 1 and 5.
Again, remember the plus/minus, giving us $\pm 1$ and $\pm 5$ .

The rational zeros theorem says that any possible rational zero is of the form $\pm \frac{\text{factor of constant term}}{\text{factor of leading coefficient}}$ .

Let's create all the combinations!
Dividing the factors of 2 ( $\pm 1$ and $\pm 2$ ) by $\pm 1$ gives us $\pm \frac{1}{1} = \pm 1$ and $\pm \frac{2}{1} = \pm 2$ .

Dividing the factors of 2 ( $\pm 1$ and $\pm 2$ ) by $\pm 5$ gives us $\pm \frac{1}{5}$ and $\pm \frac{2}{5}$ .

So, all possible rational zeros are $\pm 1$ , $\pm 2$ , $\pm \frac{1}{5}$ , and $\pm \frac{2}{5}$ .

The possible rational zeros of $f(x) = 5x^3 - 5x^2 - 4x + 2$ are $1, -1, 2, -2, \frac{1}{5}, -\frac{1}{5}, \frac{2}{5}, -\frac{2}{5}$ .