Operations on Polynomials Alıştırma Soruları — ACT — Sayfa 2

Soru 21Soru →

A computer hardware engineer is designing a rectangular silicon microchip. The total surface area of the microchip, in square millimeters, is represented by the expression $2y(3y^2 - 4y + 5)$ . A square sensor is embedded in the center of the chip, occupying an area represented by the expression $(2y - 3)^2$ square millimeters. Which of the following expressions represents the remaining surface area of the microchip, in square millimeters, that is not occupied by the sensor?

6y^3 - 12y^2 + 10y - 9

6y^3 - 12y^2 - 2y + 9

6y^3 - 12y^2 + 22y - 9

-6y^2 + 22y - 9

6y^3 - 12y^2 + 22y + 9

Cevabı ve açıklamayı göster

Cevap: $6y^3 - 12y^2 + 22y - 9$

Cevap

The correct expression is

6y^3 - 12y^2 + 22y - 9

.

To find the remaining surface area of the microchip, subtract the area of the sensor from the total area. First, expand the expression for the total area:

2y(3y^2 - 4y + 5) = 6y^3 - 8y^2 + 10y

. Next, expand the expression for the sensor area:

(2y - 3)^2 = 4y^2 - 12y + 9

. Subtract the sensor area from the total area, distributing the negative sign to all terms:

(6y^3 - 8y^2 + 10y) - (4y^2 - 12y + 9) = 6y^3 - 8y^2 + 10y - 4y^2 + 12y - 9

. Combining like terms yields the expression

6y^3 - 12y^2 + 22y - 9

.

Adım Adım Çözüm

1

Expand the total area of the microchip by distributing the monomial

2y

to each term of the trinomial

3y^2 - 4y + 5

.

2y(3y^2 - 4y + 5) = 6y^3 - 8y^2 + 10y

This determines the polynomial representing the total area before any components are subtracted.

2

Expand the area of the square sensor by squaring the binomial

(2y - 3)

.

(2y - 3)^2 = 4y^2 - 12y + 9

This determines the polynomial representing the area occupied by the sensor, ensuring the middle term

-12y

is correctly calculated.

3

Subtract the sensor's area from the total area, ensuring the negative sign is distributed to every term in the sensor's polynomial.

(6y^3 - 8y^2 + 10y) - (4y^2 - 12y + 9) = 6y^3 - 8y^2 + 10y - 4y^2 + 12y - 9

This sets up the subtraction of the two polynomials while distributing the negative sign to avoid common sign errors.

4

Combine like terms to find the final simplified expression in standard descending order.

6y^3 - 12y^2 + 22y - 9

This combines the cubic, quadratic, linear, and constant terms to find the remaining area.

Anahtar Kavram

Polynomial operations including monomial multiplication, binomial expansion, and subtraction with sign distribution.

Tahmini Süre:1m 30s

Soru 22Soru →

A rectangular garden has a length of $3x^2$ meters and a width of $2x^3 - 5$ meters. A square storage shed built inside the garden has a side length of $x^2 - 3$ meters. Which of the following polynomials represents the area of the remaining garden space, in square meters, that is not covered by the shed?

6x^5 - x^4 - 15x^2 - 9

6x^6 - x^4 - 9x^2 - 9

6x^5 - x^4 - 21x^2 + 9

6x^5 - x^4 - 9x^2 - 9

6x^5 - x^4 - 9x^2 + 9

Cevabı ve açıklamayı göster

Cevap: $6x^5 - x^4 - 9x^2 - 9$

Cevap

The polynomial

6x^5 - x^4 - 9x^2 - 9

represents the remaining area.

To find the remaining area, subtract the area of the square shed from the area of the rectangular garden. The area of the garden is

3x^2(2x^3 - 5) = 6x^5 - 15x^2

. The area of the shed is

(x^2 - 3)^2 = x^4 - 6x^2 + 9

. Subtracting the two gives

(6x^5 - 15x^2) - (x^4 - 6x^2 + 9) = 6x^5 - 15x^2 - x^4 + 6x^2 - 9 = 6x^5 - x^4 - 9x^2 - 9

, which represents the correct remaining garden space.

Adım Adım Çözüm

1

Calculate the area of the rectangular garden.

\text{Area}_{\text{garden}} = 3x^2(2x^3 - 5) = 6x^5 - 15x^2

The area of a rectangle is the product of its length and width. Applying the distributive property and the exponent rule for multiplication (

x^a \cdot x^b = x^{a+b}

), we get

3x^2 \cdot 2x^3 = 6x^5

and

3x^2 \cdot (-5) = -15x^2

.

2

Calculate the area of the square storage shed.

\text{Area}_{\text{shed}} = (x^2 - 3)^2 = x^4 - 6x^2 + 9

The area of a square is the square of its side length. We expand

(x^2 - 3)^2

using the binomial square pattern

(a - b)^2 = a^2 - 2ab + b^2

, which gives

(x^2)^2 - 2(x^2)(3) + (-3)^2 = x^4 - 6x^2 + 9

.

3

Subtract the area of the shed from the area of the garden and simplify.

\text{Remaining Area} = 6x^5 - x^4 - 9x^2 - 9

We subtract the shed's area from the garden's area:

(6x^5 - 15x^2) - (x^4 - 6x^2 + 9)

. Distributing the negative sign gives

6x^5 - 15x^2 - x^4 + 6x^2 - 9

. Combining like terms and writing in descending order yields

6x^5 - x^4 - 9x^2 - 9

.

Anahtar Kavram

Operations on Polynomials

Soru 23Soru →

A rectangular swimming pool has a length of $3x - 2$ meters and a width of $2x - 5$ meters. A concrete walkway with a uniform width of $2$ meters surrounds the pool. Which of the following expressions represents the area, in square meters, of the walkway?

20x - 12

-18x + 8

20x + 8

12x^2 - 18x + 8

12x^2 - 12

Cevabı ve açıklamayı göster

Cevap: $20x - 12$

Cevap

The expression representing the area of the walkway is

20x - 12

.

The correct answer is the expression

20x - 12

. The total area including the walkway is

(3x + 2)(2x - 1) = 6x^2 + x - 2

. The pool's area is

(3x - 2)(2x - 5) = 6x^2 - 19x + 10

. Subtracting the pool's area from the total area and distributing the negative sign to all terms gives:

(6x^2 + x - 2) - (6x^2 - 19x + 10) = 6x^2 + x - 2 - 6x^2 + 19x - 10 = 20x - 12

.

Adım Adım Çözüm

1

Determine the dimensions of the entire area including the pool and the walkway.

The total length is

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

meters, and the total width is

(2x - 5) + 2(2) = 2x - 1

meters.

Since the walkway surrounds the pool on all sides, we add twice the width of the walkway to both the length and width of the pool.

2

Calculate the total area including the walkway.

Total Area =

(3x + 2)(2x - 1) = 6x^2 - 3x + 4x - 2 = 6x^2 + x - 2

square meters.

The area is found by multiplying the total length and the total width using binomial multiplication.

3

Calculate the area of the pool itself.

Pool Area =

(3x - 2)(2x - 5) = 6x^2 - 15x - 4x + 10 = 6x^2 - 19x + 10

square meters.

The area of the pool is the product of its length and width.

4

Subtract the pool area from the total area to find the area of the walkway.

Walkway Area =

(6x^2 + x - 2) - (6x^2 - 19x + 10) = 6x^2 + x - 2 - 6x^2 + 19x - 10 = 20x - 12

square meters.

Subtracting the inner pool area from the total area yields the remaining area of the walkway surrounding it.

Anahtar Kavram

Calculating the difference between two binomial products to find the area of a border.

Soru 24Soru →

The position of a particle moving along a straight line is described by the polynomial $s(t) = (2t^2 - 3t + 4)(t + 3)$ , where $t$ represents the time in seconds and $s(t)$ represents the displacement in meters. When the polynomial is expanded and written in the standard form $at^3 + bt^2 + ct + d$ , what is the value of $b + c$ ?

Cevabı ve açıklamayı göster

Cevap: -2

Cevap

The value of

b + c

is

-2

.

Expanding the polynomial

s(t) = (2t^2 - 3t + 4)(t + 3)

yields

2t^3 + 3t^2 - 5t + 12

. Comparing this to the standard form

at^3 + bt^2 + ct + d

, we find

b = 3

and

c = -5

. Summing these coefficients gives

b + c = 3 + (-5) = -2

.

Adım Adım Çözüm

1

Multiply each term of the trinomial

2t^2 - 3t + 4

by each term of the binomial

t + 3

.

2t^3 + 6t^2 - 3t^2 - 9t + 4t + 12

Apply the distributive property to expand the product of the polynomials.

2

Combine the like terms (the

t^2

terms and the

t

terms).

2t^3 + 3t^2 - 5t + 12

Group and add the coefficients of like terms to simplify the polynomial.

3

Identify the coefficients

b

(for the

t^2

term) and

c

(for the

t

term) and find their sum.

b = 3

and

c = -5

, so

b + c = 3 + (-5) = -2

Compare the simplified polynomial to the standard form

at^3 + bt^2 + ct + d

to find the coefficients, then calculate the requested sum.

Anahtar Kavram

Operations on Polynomials

Alternatif Yöntem

Instead of expanding the whole expression, you can find the individual coefficients directly. The

t^2

term is formed by

(2t^2)(3) + (-3t)(t) = 6t^2 - 3t^2 = 3t^2

, so

b = 3

. The

t

term is formed by

(-3t)(3) + (4)(t) = -9t + 4t = -5t

, so

c = -5

. Thus,

b + c = 3 - 5 = -2

.

Tahmini Süre:1m 30s

Soru 25Soru →

A business analyst models a company's weekly net profit, $N(x)$ , in dollars, as the difference between its weekly revenue, $R(x) = (2x + 5)(30 - x)$ , and its weekly production cost, $C(x) = (x - 4)^2 + 150$ , where $x$ represents the number of items sold. When $N(x)$ is simplified and written in standard form as $ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are integers, what is the value of the coefficient $b$ ?

Cevabı ve açıklamayı göster

Cevap: 63

Cevap

The coefficient

b

is

63

.

Expanding the revenue polynomial gives

R(x) = -2x^2 + 55x + 150

and the cost polynomial gives

C(x) = x^2 - 8x + 166

. Subtracting them gives

N(x) = (-2x^2 + 55x + 150) - (x^2 - 8x + 166) = -3x^2 + 63x - 16

. The coefficient of the

x

term, which corresponds to

b

, is

63

.

Adım Adım Çözüm

1

Expand the revenue expression

R(x) = (2x + 5)(30 - x)

using polynomial multiplication.

R(x) = -2x^2 + 55x + 150

To find the polynomial representing revenue in standard form.

2

Expand the cost expression

C(x) = (x - 4)^2 + 150

using binomial squaring.

C(x) = x^2 - 8x + 166

To find the polynomial representing cost in standard form.

3

Subtract the cost polynomial from the revenue polynomial:

N(x) = R(x) - C(x)

, distributing the negative sign through all terms of the cost polynomial.

N(x) = -3x^2 + 63x - 16

To determine the net profit polynomial

N(x)

in standard form.

4

Identify the coefficient

b

of the

x

term in the standard form

ax^2 + bx + c

.

b = 63

To answer the specific question asking for the coefficient of the middle term.

Anahtar Kavram

Polynomial subtraction and expansion of algebraic expressions.

Soru 26Soru →

A homeowner has a square patio with a side length of $3x - 2$ feet. She decides to cover a portion of the patio with a rectangular outdoor rug that has dimensions $2x - 1$ feet by $x + 4$ feet. Which of the following expressions represents the area, in square feet, of the patio that remains uncovered by the rug?

7x^2 - 19x

7x^2 - 7x + 8

7x^2 - 19x + 8

9x^2 - 21x + 8

7x^2 - 5x

Cevabı ve açıklamayı göster

Cevap: $7x^2 - 19x + 8$

Cevap

7x^2 - 19x + 8

The expression representing the uncovered area is obtained by subtracting the area of the rug from the area of the patio. First, the area of the square patio is calculated as

(3x - 2)^2 = 9x^2 - 12x + 4

. Second, the area of the rug is calculated as

(2x - 1)(x + 4) = 2x^2 + 7x - 4

. Subtracting the rug's area from the patio's area requires distributing the negative sign:

(9x^2 - 12x + 4) - (2x^2 + 7x - 4) = 9x^2 - 12x + 4 - 2x^2 - 7x + 4

. Combining like terms yields

7x^2 - 19x + 8

.

Adım Adım Çözüm

1

Calculate the area of the square patio by squaring its side length.

A_{\text{patio}} = (3x - 2)^2 = 9x^2 - 12x + 4

The area of a square is equal to the square of its side length:

\text{Area} = s^2

.

2

Calculate the area of the rectangular rug by multiplying its length and width.

A_{\text{rug}} = (2x - 1)(x + 4) = 2x^2 + 8x - x - 4 = 2x^2 + 7x - 4

The area of a rectangle is equal to the product of its length and width:

\text{Area} = l \times w

.

3

Subtract the area of the rug from the area of the patio, distributing the negative sign to all terms of the rug's area.

A_{\text{uncovered}} = (9x^2 - 12x + 4) - (2x^2 + 7x - 4) = 9x^2 - 12x + 4 - 2x^2 - 7x + 4 = 7x^2 - 19x + 8

Subtracting a polynomial requires distributing the negative sign to every term inside the parentheses and then combining like terms.

Anahtar Kavram

Subtracting one polynomial from another requires distributing the negative sign to every term of the subtracted polynomial before combining like terms.

Tahmini Süre:1m 30s

Soru 27Soru →

A landscape architect designs a square courtyard with a side length of $4x - 3$ meters. A square garden bed with a side length of $2x - 1$ meters is placed in one of the corners of the courtyard. The remaining area of the courtyard is paved. If the paved area, in square meters, is represented by the polynomial $ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants, what is the value of $b$ ?

Cevabı ve açıklamayı göster

Cevap: -20

Cevap

The value of

b

is

-20

.

The remaining paved area is the difference between the area of the courtyard and the garden bed:

(16x^2 - 24x + 9) - (4x^2 - 4x + 1) = 12x^2 - 20x + 8

. The coefficient of the linear

x

term is

-20

.

Adım Adım Çözüm

1

Find the polynomial representing the total area of the courtyard.

A_{\text{total}} = (4x - 3)^2 = 16x^2 - 24x + 9

square meters

The area of a square is the square of its side length. We expand

(4x - 3)^2

using the identity

(p - q)^2 = p^2 - 2pq + q^2

.

2

Find the polynomial representing the area of the garden bed.

A_{\text{garden}} = (2x - 1)^2 = 4x^2 - 4x + 1

square meters

The garden bed is also a square, so we expand

(2x - 1)^2

using the binomial squaring identity.

3

Subtract the garden bed area from the total area to find the paved area.

A_{\text{paved}} = 12x^2 - 20x + 8

square meters

The paved area is the difference between the two areas. We distribute the negative sign to all terms of the subtracted polynomial:

-(4x^2 - 4x + 1) = -4x^2 + 4x - 1

, and then combine like terms.

4

Identify the coefficient of the

x

term,

b

.

b = -20

Comparing

12x^2 - 20x + 8

to

ax^2 + bx + c

, the coefficient of the linear

x

term is

-20

.

Anahtar Kavram

Operations on Polynomials

Alternatif Yöntem

To find only the coefficient of the linear term, expand and subtract only the linear terms from both binomial expansions:

2(4x)(-3) - 2(2x)(-1) = -24x - (-4x) = -20x

. The coefficient

b

is therefore

-20

.

Tahmini Süre:1m 30s

Soru 28Soru →

A square metal sheet has a side length of $3x^3$ inches. A square cutout with a side length of $(x^3 - 2)$ inches is removed from the center of the sheet. Which of the following polynomial expressions represents the area, in square inches, of the remaining metal sheet?

8x^6 - 4x^3 + 4

8x^5 + 4x^3 - 4

8x^6 + 4

8x^6 + 4x^3 - 4

2x^6 + 4x^3 - 4

Cevabı ve açıklamayı göster

Cevap: $8x^6 + 4x^3 - 4$

Cevap

The expression

8x^6 + 4x^3 - 4

represents the remaining area of the metal sheet.

To find the remaining area, subtract the area of the square cutout from the total area of the metal sheet. The total area is the square of the side length

3x^3

, which is

(3x^3)^2 = 9x^6

. The area of the cutout is the square of

(x^3 - 2)

, which is

(x^3 - 2)^2 = x^6 - 4x^3 + 4

. Subtracting the cutout area from the total area yields

9x^6 - (x^6 - 4x^3 + 4) = 9x^6 - x^6 + 4x^3 - 4 = 8x^6 + 4x^3 - 4

. This matches the correct expression.

Adım Adım Çözüm

1

Calculate the total area of the square metal sheet.

Area_{total} = (3x^3)^2 = 9x^6

The area of a square is the square of its side length. According to the power of a product rule,

(ab)^n = a^n b^n

, and the power of a power rule,

(x^a)^b = x^{ab}

.

2

Calculate the area of the square cutout.

Area_{cutout} = (x^3 - 2)^2 = x^6 - 4x^3 + 4

Squaring the binomial yields

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

, which simplifies to

x^6 - 4x^3 + 4

.

3

Subtract the area of the cutout from the total area of the metal sheet.

Area_{remaining} = 9x^6 - (x^6 - 4x^3 + 4) = 8x^6 + 4x^3 - 4

Distributing the negative sign through the parentheses yields

9x^6 - x^6 + 4x^3 - 4

. Combining the like terms

9x^6

and

-x^6

gives the final simplified expression.

Anahtar Kavram

Polynomial subtraction and binomial expansion

Tahmini Süre:1m 30s

Soru 29Soru →

The polynomial $4x^2 - 8x - 5$ is subtracted from the polynomial $7x^2 - 3x + 4$ . The simplified difference can be expressed as $ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constant integers. What is the value of the coefficient $b$ ?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The coefficient of the

x

term,

b

, is 5.

Subtracting

4x^2 - 8x - 5

from

7x^2 - 3x + 4

yields

(7x^2 - 4x^2) + (-3x - (-8x)) + (4 - (-5)) = 3x^2 + 5x + 9

. The coefficient of the linear term

x

is

5

.

Adım Adım Çözüm

1

Set up the subtraction expression.

(7x^2 - 3x + 4) - (4x^2 - 8x - 5)

Subtracting the second polynomial from the first requires enclosing the second polynomial in parentheses to apply the subtraction to all terms.

2

Distribute the negative sign to all terms inside the parentheses.

7x^2 - 3x + 4 - 4x^2 + 8x + 5

Distributing the subtraction sign flips the sign of each term: positive terms become negative, and negative terms become positive.

3

Group and combine like terms.

3x^2 + 5x + 9

Combine the coefficients of matching variable parts:

(7 - 4)x^2 = 3x^2

,

(-3 + 8)x = 5x

, and

4 - (-5) = 9

.

4

Identify the coefficient

b

corresponding to the

x

term.

b = 5

Comparing the simplified expression

3x^2 + 5x + 9

to

ax^2 + bx + c

shows that the coefficient of

x

is 5.

Anahtar Kavram

Polynomial Subtraction and Combining Like Terms

Soru 30Soru →

An industrial designer is drafting a template for a rectangular solar panel. The total area of the panel is represented by the polynomial $4x^3(3x^2 - 2x + 5)$ square centimeters. A rectangular sensor cutout with an area of $(2x^2 - 3)^2$ square centimeters is removed from the panel. Which of the following polynomials represents the remaining area, in square centimeters, of the solar panel in terms of $x$ ?

12x^5 - 12x^4 + 20x^3 - 12x^2 + 9

12x^5 - 12x^4 + 20x^3 - 9

12x^5 - 12x^4 + 20x^3 + 12x^2 - 9

12x^6 - 4x^4 + 12x^3 + 12x^2 - 9

12x^5 - 12x^4 + 20x^3 + 12x^2 + 9

Cevabı ve açıklamayı göster

Cevap: $12x^5 - 12x^4 + 20x^3 + 12x^2 - 9$

Cevap

12x^5 - 12x^4 + 20x^3 + 12x^2 - 9

To find the remaining area, calculate the total area and subtract the cutout area. The total area is

4x^3(3x^2 - 2x + 5) = 12x^5 - 8x^4 + 20x^3

. The area of the cutout is

(2x^2 - 3)^2 = 4x^4 - 12x^2 + 9

. Subtracting the cutout polynomial requires distributing the negative sign to all three terms:

(12x^5 - 8x^4 + 20x^3) - (4x^4 - 12x^2 + 9) = 12x^5 - 8x^4 + 20x^3 - 4x^4 + 12x^2 - 9

. Combining like terms yields the expression

12x^5 - 12x^4 + 20x^3 + 12x^2 - 9

.

Adım Adım Çözüm

1

Find the polynomial representing the total area of the solar panel by distributing

4x^3

through the expression

(3x^2 - 2x + 5)

.

12x^5 - 8x^4 + 20x^3

When multiplying terms with the same base, add their exponents (

x^a \cdot x^b = x^{a+b}

).

2

Find the polynomial representing the area of the sensor cutout by expanding

(2x^2 - 3)^2

.

4x^4 - 12x^2 + 9

Use the binomial squaring identity

(a - b)^2 = a^2 - 2ab + b^2

, where

a = 2x^2

and

b = 3

.

3

Subtract the cutout area from the total area, ensuring the negative sign is distributed to every term in the cutout polynomial.

(12x^5 - 8x^4 + 20x^3) - (4x^4 - 12x^2 + 9) = 12x^5 - 12x^4 + 20x^3 + 12x^2 - 9

Distributing the negative sign changes the signs of all terms in the subtracted polynomial, allowing like terms to be combined.

Anahtar Kavram

Polynomial subtraction and expansion using binomial squaring and exponent properties

Tahmini Süre:1m 30s

Operations on Polynomials

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

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Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

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Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Alternatif Yöntem

Cevap

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Anahtar Kavram

Sık Yapılan Hatalar

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Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Alternatif Yöntem

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar