Factor the polynomial completely. Check your, №18009982, 23.06.2021 12:40

Mathematics

Factor the polynomial completely. Check your answer. 20x^3 - 245xy^2

Step-by-step answer

P Answered by Specialist

5x(4x^2 - 49y^2) or 5x(2x-7y)(2x+7y)

Step-by-step explanation:

20x^3 - 245xy^2

5x(4x^2 - 49y^2)

Check your

5x(4x^2 - 49y^2)

5x(4x^2) + 5x(-49y^2)

20x^3-245xy^2

5x(4x^2 - 49y^2) can also be written as:

5x(4x^2 - 49y^2)

5x(2x-7y)(2x+7y)

5x(4x^2+14xy-14xy+49y^2)

5x(4x^2+49y^2)

5x(4x^2) + 5x(-49y^2)

20x^3-245xy^2

Hope this helps!

Mathematics

Prompt: his interactive animation shows an image of a brick patio and a garden and grass area. the title patio project is shown and a begin button starts the animation. [click begin. ] on-screen text and audio: patio project: you have a small patio and want to improve it. you can build one of the following features: a fish pond, a planter box, or a fire pit. choose one of the projects, and drag it to the patio. [an image of an orange fish in water, a planter box with flowers, and a fire pit can be seen on the left side of the screen. any of the

Step-by-step answer

P Answered by PhD

341

For this answer let's pick the Fish Pond.

QUESTION: What do you know?
We know that the total area of the rectangular patio is 24 square feet. The rectangular fish pond that we want to build should have the length that is twice the width. We also need a 1-foot border around the pond.

The total area of the patio is represented by $2 x^{2} +6x+4$ where x is the width of the fish pond.

QUESTION: What do you want to find out?
We can answer this question by examining what we have been given (we have answered this in the previous question). Since we only have one variable in the equation and this is the width of the fish pond, then this is the only thing that we need to find out. The length will just follow after the width is solved. (Basically, we are concerned with the dimensions of our project.)

QUESTION: What kind of answer do you expect?
Since we are only expecting to solve for the dimensions, then we should expect two numbers which would represent the width and the length. Through the polynomial, we can get the width of our project, then the length would just be twice our answer for the width.

1. To set our polynomial equation equal to the total area of our patio, we just write the equation on the left side and the total area of the patio on the right. We recall the equation that we gave in the first question above as well as the patio's area.

$2x^{2} +6x+4=24$

2. For this item, we just simply negate the constant term on the right side by subtracting the same number to both sides. Since our constant term is 24, we subtract 24 from the polynomial as well as the term on the right side.

$2x^{2} +6x+4-24=24-24$
$2x^{2} +6x-20=0$

3. In this item we are only tasked to find the GCF of the trinomial and factor it out of the left side of the equation. The GCF just means the common number or variable in all terms. Upon examining, we can see that the only common factor of the terms is 2 (since all terms are divisible by 2), thus this will be the GCF and we will factor this one out.

$2(x^{2} +3x-10)=0$

4. To factor the polynomial completely, we just figure out the possible factors of the quadratic equation. You can do this by trying out all factors of -10 that will have a sum of 3 (since -10 is the constant and 3 is the coefficient of x). Upon examining you should end up with:

2(x+5)(x-2)=0

(Notice that 5 and -2 are factors of -10 and their sum is 3).

5. The rule stated on this item just means that we can equate each individual factor to zero (except for the constant term) to find out the possible values of x. That means that we can solve for x by using the equations x+5=0 and x-2=0.

x+5=0

6. For this item, we just do simple substitutions to the equation $2x^{2} +6x-20=0$ to verify whether the values we got in question 5 is really a solution to the equation. The values of x that we got are -5 and 2 so we substitute these one at a time.

$2(-5)^{2} +6(-5)-20=0$
50-30-20=0

$2(2)^{2} +6(2)-20=0$
8+12-20=0

Both values of x make the left side equal to zero therefore both are solutions to the equation.

7. We can tell the dimensions of the project by looking at the values that we got for x, since we assumed x to be the width of our project. The fact that we got two values for x won't be a problem since one of these values is negative. There is no negative measurement/width so we can just ignore this negative value (-5). Thus, the width for our project would be 2 and the length would be twice this value which is 4.

Width = 2 feet
Length = 4 feet

8. For this item, we just illustrate the dimensions of our project and add a 1-foot border (as we are told initially). You can see this illustration in the picture I attached below.

Since we have an extra 1 foot in every side, that means we need to add 2 feet to both the width and the length. Therefore, our width now is 4 while our length is 6.

Multiplying these numbers to verify the area, we get 24 square feet which is exactly the area of the patio.
Prompt: his interactive animation shows an image of a brick patio and a garden and grass area. the

Mathematics

Prompt: his interactive animation shows an image of a brick patio and a garden and grass area. the title patio project is shown and a begin button starts the animation. [click begin. ] on-screen text and audio: patio project: you have a small patio and want to improve it. you can build one of the following features: a fish pond, a planter box, or a fire pit. choose one of the projects, and drag it to the patio. [an image of an orange fish in water, a planter box with flowers, and a fire pit can be seen on the left side of the screen. any of the

Step-by-step answer

P Answered by PhD

341

For this answer let's pick the Fish Pond.

QUESTION: What do you know?
We know that the total area of the rectangular patio is 24 square feet. The rectangular fish pond that we want to build should have the length that is twice the width. We also need a 1-foot border around the pond.

The total area of the patio is represented by $2 x^{2} +6x+4$ where x is the width of the fish pond.

QUESTION: What do you want to find out?
We can answer this question by examining what we have been given (we have answered this in the previous question). Since we only have one variable in the equation and this is the width of the fish pond, then this is the only thing that we need to find out. The length will just follow after the width is solved. (Basically, we are concerned with the dimensions of our project.)

QUESTION: What kind of answer do you expect?
Since we are only expecting to solve for the dimensions, then we should expect two numbers which would represent the width and the length. Through the polynomial, we can get the width of our project, then the length would just be twice our answer for the width.

1. To set our polynomial equation equal to the total area of our patio, we just write the equation on the left side and the total area of the patio on the right. We recall the equation that we gave in the first question above as well as the patio's area.

$2x^{2} +6x+4=24$

2. For this item, we just simply negate the constant term on the right side by subtracting the same number to both sides. Since our constant term is 24, we subtract 24 from the polynomial as well as the term on the right side.

$2x^{2} +6x+4-24=24-24$
$2x^{2} +6x-20=0$

3. In this item we are only tasked to find the GCF of the trinomial and factor it out of the left side of the equation. The GCF just means the common number or variable in all terms. Upon examining, we can see that the only common factor of the terms is 2 (since all terms are divisible by 2), thus this will be the GCF and we will factor this one out.

$2(x^{2} +3x-10)=0$

4. To factor the polynomial completely, we just figure out the possible factors of the quadratic equation. You can do this by trying out all factors of -10 that will have a sum of 3 (since -10 is the constant and 3 is the coefficient of x). Upon examining you should end up with:

2(x+5)(x-2)=0

(Notice that 5 and -2 are factors of -10 and their sum is 3).

5. The rule stated on this item just means that we can equate each individual factor to zero (except for the constant term) to find out the possible values of x. That means that we can solve for x by using the equations x+5=0 and x-2=0.

x+5=0

6. For this item, we just do simple substitutions to the equation $2x^{2} +6x-20=0$ to verify whether the values we got in question 5 is really a solution to the equation. The values of x that we got are -5 and 2 so we substitute these one at a time.

$2(-5)^{2} +6(-5)-20=0$
50-30-20=0

$2(2)^{2} +6(2)-20=0$
8+12-20=0

Both values of x make the left side equal to zero therefore both are solutions to the equation.

7. We can tell the dimensions of the project by looking at the values that we got for x, since we assumed x to be the width of our project. The fact that we got two values for x won't be a problem since one of these values is negative. There is no negative measurement/width so we can just ignore this negative value (-5). Thus, the width for our project would be 2 and the length would be twice this value which is 4.

Width = 2 feet
Length = 4 feet

8. For this item, we just illustrate the dimensions of our project and add a 1-foot border (as we are told initially). You can see this illustration in the picture I attached below.

Since we have an extra 1 foot in every side, that means we need to add 2 feet to both the width and the length. Therefore, our width now is 4 while our length is 6.

Multiplying these numbers to verify the area, we get 24 square feet which is exactly the area of the patio.
Prompt: his interactive animation shows an image of a brick patio and a garden and grass area. the

Mathematics

Factor the polynomial F(x) =x^3-x^2-4x+4 completely. Part 1: Find and list all the possible roots of F(x). Part 2: Use the remainder Theorem to determine which of the roots from Part I are roots of F(x). Part 3:Factor the polynomial F(x)=x^3-x^2-4x+4 completely show your work. Part 4:check your answer from Part 3 by multiplying the factors. Show your work

Step-by-step answer

P Answered by Specialist

19

Step-by-step explanation:

PART 1:

Possible roots by noticing the coefficient of first term:

x = 1, -1

PART 2:

1 | 1 -1 -4 4

1 0 -4

1 0 -4 0

The remainder is zero, hence one factor is (x-1)

PART 3:

$x^3-x^2-4x+4=(x^2-4)(x-1)\\\\x^3-x^2-4x+4=(x+2)(x-2)(x-1)$

PART 4:

$f(x)=(x+2)(x-2)(x-1)\\\\=(x^2-4)(x-1)\\\\=x^3-4x-x^2+4\\\\=x^3-x^2-4x+4$

Mathematics

Factor the polynomial F(x) =x^3-x^2-4x+4 completely. Part 1: Find and list all the possible roots of F(x). Part 2: Use the remainder Theorem to determine which of the roots from Part I are roots of F(x). Part 3:Factor the polynomial F(x)=x^3-x^2-4x+4 completely show your work. Part 4:check your answer from Part 3 by multiplying the factors. Show your work

Step-by-step answer

P Answered by Master

19

Step-by-step explanation:

PART 1:

Possible roots by noticing the coefficient of first term:

x = 1, -1

PART 2:

1 | 1 -1 -4 4

1 0 -4

1 0 -4 0

The remainder is zero, hence one factor is (x-1)

PART 3:

$x^3-x^2-4x+4=(x^2-4)(x-1)\\\\x^3-x^2-4x+4=(x+2)(x-2)(x-1)$

PART 4:

$f(x)=(x+2)(x-2)(x-1)\\\\=(x^2-4)(x-1)\\\\=x^3-4x-x^2+4\\\\=x^3-x^2-4x+4$

Mathematics

Complete the sentence below. if (3x2 + 22x + 7) ÷(x + 7) = 3x + 1, then (x + = . the check of the polynomial division problem shows that the product of two polynomials is a polynomial. this supports the fact that the property is satisfied for polynomial multiplication.

Step-by-step answer

P Answered by Master

The answer is (x+7)(3x+1)=33x^2+22x+7.

Step-by-step explanation: Given that

$(3x2 + 22x + 7) \div(x + 7) = 3x + 1,$

and we are to complete the following sentence:

(x+7)(???)=???

We have the following division algorithm for polynomials

$\textup{If }a(x)\times b(x)=c(x),\\\textup{then, we have }\\\\\dfrac{c(x)}{b(x)}=a(x)~~~~~\textup{or}~~~~~~c(x)\div b(x)=a(x).$

Here, a(x) = quotient, b(x) = divisor and c(x) = dividend.

Applying this rule in the given problem, we have

$\textup{since }(3x2 + 22x + 7) \div(x + 7) = 3x + 1,\\\\\textup{so, }\\\\(x+7)(3x+1)=3x^2+22x+7=0.$

Thus, the complete sentence is

(x+7)(3x+1)=3x^2+22x+7=0.

Mathematics

1. check all polynomials that are perfect square trinomials: 9x2 − 6x + 1 36b2 − 24b + 8 16x2 + 24x + 9 4a2 − 10a + 25 2. factor completely: x2 − 8x + 16 (x + 4)(x + 4) (x − 4)(x − 4) (x + 4)(x − 4) (x − 2)(x − 8) 3. factor completely: 16x2 + 40x + 25 (4x − 5)(4x − 5) (2x − 5)(2x − 5) (2x + 5)(2x + 5) (4x + 5)(4x + 5) 4. rhett decides to build a square room for his movie and music collection. if the area of the room is 9x2 − 6x + 1 square feet, what is the length of one side of the room? (3x + 1) feet (9x − 1) feet (3x − 1) feet (9x + 1) feet 5.

Step-by-step answer

P Answered by Master

26

1. Check all polynomials that are perfect square trinomials:9x2 − 6x + 1
16x2 + 24x + 9

2. Factor completely: x2 − 8x + 16

(x − 4)(x − 4)

3. Factor completely: 16x2 + 40x + 25

(4x + 5)(4x + 5)

4. Rhett decides to build a square room for his movie and music collection. If the area of the room is 9x^2 − 6x + 1 square feet, what is the length of one side of the room?

(3x − 1) feet

5. Given the following perfect square trinomial, fill in the missing term. (Do not type the variable in the blank.)

4x2 + ___x + 49

28

Mathematics

Jay factored the 4 term polynomial x^3-9x : + : 2x^2-18 x 3 − 9 x + 2 x 2 − 18 and decided that the complete factorization was (x+2 right) ( x + 2 ) left(x^2-9 right) ( x 2 − 9 ) . before turning in his paper, he checked his final factorization by multiplying out his factors and was sure that he had found the correct factors. when his teacher graded his paper, she marked his answer as incorrect, but gave him one more chance to show the correct factorization. a) what was jay s mistake? b) show/describe each step in factoring the 4 term expression

Step-by-step answer

P Answered by PhD

Correct factorization: (x+2)(x+3)(x-3)

Step-by-step explanation:

The given 4 term polynomial is:

$x^{3}-9x+2x^{2}-18$

Part a) Jay's Mistake:

Factorization of Jay was:

$(x+2)(x^{2}-9)$

Though this expression will simplify to original given expression but this is not the complete and final factorization. The second factor which is x² - 9 can be factored further, which is shown in the next part.

Part b) Complete Factorization

In order to factor a 4 term expression of the type given in the question, the first step is to take the common from similar terms. You might need to re-arrange the terms before taking common in some case. Taking commons from the given expression, we get:

$x^{3}-9x+2x^{2}-18\\\\ = x(x^{2}-9)+2(x^{2}-9)\\\\ =(x+2)(x^{2}-9)$

Jay stopped at this step. At this step you need to look if any part of the expression can be factored further. Luckily, in this case x² - 9 can be factored further as its a difference of perfect squares:

x² - 9 = x² - (3)² = (x + 3)(x - 3)

Using these factors of x² - 9 in previous expression, we get:

$(x+2)(x^{2}-9)\\\\ = (x+2)(x+3)(x-3)$

This is the final factored form of the given 4 term expression as it can not be factored further in any way.

Mathematics

Ireally need with this show work jay factored the 4 term polynomial x^3 - 9x + 2x^2 - 18 and decided that the complete factorization was (x+2) (x^2 - 9) before turning in his paper, he checked his final factorization by multiplying out his factors and was sure that he had found the correct factors. when his teacher graded his paper, she marked his answer as incorrect, but gave him one more chance to show the correct factorization. a) what was jay s mistake? b) show/describe each step in factoring the 4 term expression correctly and completely:

Step-by-step answer

P Answered by PhD

1

x^3 - 9x + 2x^2 - 18

It's a bit odd to write the polynomial this way. Let's sort by degree.

x^3 + 2x^2 -9x - 18

That's less confusing. Let's check Jay's factorization:

(x+2) (x^2 - 9) = x^3 + 2x^2 - 9x - 18

Seems correct.

(a) Jay's problem is that he still has more factoring to do. The second factor he has is the difference of two squares:

(x+2) (x^2 - 9) = (x+2)(x - 3)(x + 3)

(b)

Let's factor f(x) = x^3 - 9x + 2x^2 - 18

Playing around with some small numbers we find

f(-2) = 0

so (x + 2) is a factor.

x^3 + 2x^2 -9x - 18

= x^2(x + 2) -9(x + 2)

= (x^2 - 9)(x+ 2)

= (x+3)(x-3)(x+2)

Mathematics

Jay factored the 4 term polynomial x^3-9x : + : 2x^2-18 x 3 − 9 x + 2 x 2 − 18 and decided that the complete factorization was (x+2 right) ( x + 2 ) left(x^2-9 right) ( x 2 − 9 ) . before turning in his paper, he checked his final factorization by multiplying out his factors and was sure that he had found the correct factors. when his teacher graded his paper, she marked his answer as incorrect, but gave him one more chance to show the correct factorization. a) what was jay s mistake? b) show/describe each step in factoring the 4 term expression

Step-by-step answer

P Answered by PhD

Correct factorization: (x+2)(x+3)(x-3)

Step-by-step explanation:

The given 4 term polynomial is:

$x^{3}-9x+2x^{2}-18$

Part a) Jay's Mistake:

Factorization of Jay was:

$(x+2)(x^{2}-9)$

Though this expression will simplify to original given expression but this is not the complete and final factorization. The second factor which is x² - 9 can be factored further, which is shown in the next part.

Part b) Complete Factorization

In order to factor a 4 term expression of the type given in the question, the first step is to take the common from similar terms. You might need to re-arrange the terms before taking common in some case. Taking commons from the given expression, we get:

$x^{3}-9x+2x^{2}-18\\\\ = x(x^{2}-9)+2(x^{2}-9)\\\\ =(x+2)(x^{2}-9)$

Jay stopped at this step. At this step you need to look if any part of the expression can be factored further. Luckily, in this case x² - 9 can be factored further as its a difference of perfect squares:

x² - 9 = x² - (3)² = (x + 3)(x - 3)

Using these factors of x² - 9 in previous expression, we get:

$(x+2)(x^{2}-9)\\\\ = (x+2)(x+3)(x-3)$

This is the final factored form of the given 4 term expression as it can not be factored further in any way.

Factor the polynomial completely. Check your answer.
20x^3 - 245xy^2

Step-by-step answer

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