Multiplying Polynomials Alıştırma Soruları — Armed Services Vocational Aptitude Battery (ASVAB) — Sayfa 2

Soru 21Soru →

If the polynomial expression $(3x - 5)$ is multiplied by the expression $(Ax^2 + 2x - 7)$ , the resulting simplified polynomial has an $x^2$ coefficient of $-19$ . What is the value of the constant $A$ ?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

5

The

x^2

terms in the product of

(3x - 5)

and

(Ax^2 + 2x - 7)

are generated by multiplying the

x

terms together

(3x \cdot 2x = 6x^2)

and by multiplying the constant by the

x^2

term

(-5 \cdot Ax^2 = -5Ax^2)

. Setting their sum equal to

-19x^2

gives the equation

6 - 5A = -19

. Solving for

A

yields

A = 5

.

Adım Adım Çözüm

1

Identify combinations that yield an

x^2

term

The

x^2

terms come from multiplying

(3x)

by

(2x)

and

(-5)

by

(Ax^2)

.

To find the coefficient of a specific term without expanding the entire polynomial, only multiply the terms whose exponents sum to the target power.

2

Calculate the partial products

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

and

(-5)(Ax^2) = -5Ax^2

These represent the only contributions to the

x^2

term in the final expanded polynomial.

3

Formulate an equation for the coefficient

6 - 5A = -19

The sum of the partial

x^2

coefficients must equal the total

x^2

coefficient given in the problem.

4

Solve the linear equation for

A

-5A = -25 \rightarrow A = 5

Isolating the variable

A

provides the final answer.

Anahtar Kavram

Multiplying Polynomials

Soru 22Soru →

A polynomial $P(x)$ is defined as the product of the binomial $(4x - 3)$ and the trinomial $(-2x^2 + 5x - 1)$ . What is the fully expanded and simplified form of $P(x)$ ?

-8x^3 + 14x^2 - 19x - 3

-2x^2 + x + 3

-8x^3 + 26x^2 - 19x + 3

-8x^3 - 15x + 3

Cevabı ve açıklamayı göster

Cevap: $-8x^3 + 26x^2 - 19x + 3$

Cevap

-8x^3 + 26x^2 - 19x + 3

The correct expression is found by successfully distributing both

4x

and

-3

across the trinomial

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

, maintaining correct exponents and signs, and finally combining the like terms:

20x^2

with

6x^2

, and

-4x

with

-15x

.

Adım Adım Çözüm

1

Set up the multiplication using the distributive property.

4x(-2x^2 + 5x - 1) - 3(-2x^2 + 5x - 1)

Every term in the first polynomial must be multiplied by every term in the second polynomial.

2

Distribute

4x

through the trinomial.

4x(-2x^2) + 4x(5x) + 4x(-1) = -8x^3 + 20x^2 - 4x

Multiply coefficients and add exponents for the variable base.

3

Distribute

-3

through the trinomial.

-3(-2x^2) - 3(5x) - 3(-1) = 6x^2 - 15x + 3

Multiply the constant by each term, carefully applying integer sign rules.

4

Combine the results and group like terms.

-8x^3 + (20x^2 + 6x^2) + (-4x - 15x) + 3

Grouping terms with the same variable exponent prepares the expression for final simplification.

5

Add the coefficients of the like terms.

-8x^3 + 26x^2 - 19x + 3

This yields the polynomial in standard form.

Anahtar Kavram

Multiplying polynomials by distributing every term in the first factor to every term in the second, then combining like terms.

Soru 23Soru →

A manufacturer is cutting a square opening out of a rectangular metal panel. The rectangular panel has a length of $(4x - 1)$ inches and a width of $(2x + 3)$ inches. The square opening to be removed has a side length of $(x + 2)$ inches. Which of the following expressions represents the total remaining area of the metal panel after the square is removed?

7x^2 + 6x - 7

7x^2 + 10x - 7

7x^2 + 14x + 1

7x^2 - 7

Cevabı ve açıklamayı göster

Cevap: $7x^2 + 6x - 7$

Cevap

The correct expression is

7x^2 + 6x - 7

.

The remaining area is found by subtracting the square's area from the rectangle's area. Multiplying the dimensions

(4x - 1)(2x + 3)

yields

8x^2 + 10x - 3

. Squaring the dimension

(x + 2)

yields

x^2 + 4x + 4

. Subtracting the entire second polynomial requires distributing the negative sign, resulting in

8x^2 + 10x - 3 - x^2 - 4x - 4

, which correctly simplifies to

7x^2 + 6x - 7

.

Adım Adım Çözüm

1

Calculate the area of the rectangular panel.

(4x - 1)(2x + 3) = 8x^2 + 12x - 2x - 3 = 8x^2 + 10x - 3

The area of a rectangle is length multiplied by width, which requires multiplying two binomials using the FOIL method.

2

Calculate the area of the square opening.

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

The area of a square is its side length multiplied by itself.

3

Subtract the square's area from the rectangle's area.

(8x^2 + 10x - 3) - (x^2 + 4x + 4) = 8x^2 + 10x - 3 - x^2 - 4x - 4 = 7x^2 + 6x - 7

To find the remaining area, the removed portion must be subtracted. It is critical to distribute the negative sign to every term inside the parentheses being subtracted.

Anahtar Kavram

Multiplying binomials and subtracting polynomials while properly distributing negative signs.

Soru 24Soru →

Which of the following expressions is equivalent to the product of $(5x - 2)$ and $(x^2 - 3x - 4)$ ?

5x^3 + 8

5x^3 - 17x^2 - 26x - 8

5x^3 - 17x^2 - 14x + 8

5x^3 - 31x + 8

Cevabı ve açıklamayı göster

Cevap: $5x^3 - 17x^2 - 14x + 8$

Cevap

The correct answer is

5x^3 - 17x^2 - 14x + 8

.

The correct expression is found by systematically distributing each term in the first polynomial to each term in the second. First, multiplying

5x

by

(x^2 - 3x - 4)

yields

5x^3 - 15x^2 - 20x

. Next, multiplying

-2

by

(x^2 - 3x - 4)

yields

-2x^2 + 6x + 8

. Finally, combining the like terms (

-15x^2

and

-2x^2

becomes

-17x^2

;

-20x

and

6x

becomes

-14x

) results in the simplified polynomial

5x^3 - 17x^2 - 14x + 8

.

Adım Adım Çözüm

1

Distribute the first term of the binomial,

5x

, to each term in the trinomial.

5x(x^2 - 3x - 4) = 5x^3 - 15x^2 - 20x

To expand the polynomials, every term in the first polynomial must be multiplied by every term in the second.

2

Distribute the second term of the binomial,

-2

, to each term in the trinomial.

-2(x^2 - 3x - 4) = -2x^2 + 6x + 8

The second term, along with its negative sign, must also be multiplied by every term in the trinomial.

3

Add the results from Step 1 and Step 2 and combine like terms.

(5x^3 - 15x^2 - 20x) + (-2x^2 + 6x + 8) = 5x^3 - 17x^2 - 14x + 8

Combining terms with the same variable to the same power fully simplifies the resulting polynomial.

Anahtar Kavram

Multiplying Polynomials using Distribution

Soru 25Soru →

A rectangular garden has a length of $(3x - 2)$ meters and a width of $(2x + 5)$ meters. Inside the garden, a smaller rectangular paved patio is built with a length of $(x + 4)$ meters and a width of $2x$ meters. Which of the following expressions represents the total area of the garden that is NOT paved, in square meters?

4x^2 + 3x - 10

4x^2 + 19x - 10

4x^2 - 8x - 10

4x^2 - 7x

Cevabı ve açıklamayı göster

Cevap: $4x^2 + 3x - 10$

Cevap

The expression representing the unpaved area is

4x^2 + 3x - 10

.

The correct answer is found by calculating the area of the entire garden and subtracting the area of the patio. First, multiply the dimensions of the garden using the distributive property (FOIL):

(3x - 2)(2x + 5) = 6x^2 + 15x - 4x - 10 = 6x^2 + 11x - 10

. Next, find the area of the patio by distributing

2x

:

(x + 4)(2x) = 2x^2 + 8x

. Finally, subtract the patio's area from the total area:

(6x^2 + 11x - 10) - (2x^2 + 8x)

. Make sure to distribute the negative sign to both terms in the second polynomial:

6x^2 + 11x - 10 - 2x^2 - 8x = 4x^2 + 3x - 10

.

Adım Adım Çözüm

1

Calculate the total area of the garden by multiplying its length and width.

(3x - 2)(2x + 5) = 6x^2 + 15x - 4x - 10 = 6x^2 + 11x - 10

The area of a rectangle is found by multiplying its length by its width.

2

Calculate the area of the paved patio by multiplying its length and width.

(x + 4)(2x) = 2x^2 + 8x

We need the area of the inner rectangle in order to subtract it from the total area.

3

Subtract the patio's area from the total garden area to find the unpaved area.

(6x^2 + 11x - 10) - (2x^2 + 8x) = 6x^2 + 11x - 10 - 2x^2 - 8x = 4x^2 + 3x - 10

The unpaved region is the difference between the total area and the paved area. Note that the negative sign must be distributed to both terms in the second polynomial.

Anahtar Kavram

Multiplying Polynomials

Soru 26Soru →

Determine the value of the missing coefficient by evaluating the problem below.

Aşağıdaki boşlukları doldurun

A rectangular solar panel has a length of

(5k - 4)

inches and a width of

(3k + 7)

inches. The area of the panel is represented by the fully expanded polynomial

Ak^2 + Bk + C

square inches. The value of the coefficient

B

is .

Cevabı ve açıklamayı göster

Cevap

23

By multiplying the binomials

(5k - 4)

and

(3k + 7)

, the expanded expression is

15k^2 + 35k - 12k - 28

. Combining the middle terms (

35k - 12k

) yields

23k

. Since the polynomial is in the standard form

Ak^2 + Bk + C

, the value of

B

is 23.

Adım Adım Çözüm

1

Set up the area calculation by multiplying the length and width binomials.

Area =

(5k - 4)(3k + 7)

The area of a rectangle is found by multiplying its length by its width.

2

Distribute the terms using the FOIL method (First, Outer, Inner, Last).

15k^2 + 35k - 12k - 28

Each term in the first binomial must be multiplied by each term in the second binomial.

3

Combine the like terms in the middle to find the value of

B

.

15k^2 + 23k - 28

Adding

35k

and

-12k

simplifies the polynomial to the standard form

Ak^2 + Bk + C

.

4

Identify the coefficient

B

corresponding to the

k

term.

B = 23

The polynomial is

15k^2 + 23k - 28

, so the coefficient of the

k

term is 23.

Anahtar Kavram

Multiplying binomials and combining like terms to find a specific coefficient.

Soru 27Soru →

Which of the following represents the fully simplified product of the polynomials $(3a - 2b)$ and $(4a^2 + 5ab - b^2)$ ?

12a^3 + 7a^2b - 13ab^2 + 2b^3

12a^3 + 2b^3

12a^3 + 23a^2b + 7ab^2 - 2b^3

12a^3 - 6a^2b^2 + 2b^3

Cevabı ve açıklamayı göster

Cevap: $12a^3 + 7a^2b - 13ab^2 + 2b^3$

Cevap

The fully simplified product is

12a^3 + 7a^2b - 13ab^2 + 2b^3

.

The correct answer is obtained by distributing each term in the binomial to every term in the trinomial, and then combining the like terms. Multiplying

3a

by each term yields

12a^3 + 15a^2b - 3ab^2

. Multiplying

-2b

by each term yields

-8a^2b - 10ab^2 + 2b^3

. Adding these together and combining the like terms (

15a^2b - 8a^2b = 7a^2b

and

-3ab^2 - 10ab^2 = -13ab^2

) results in the final expression.

Adım Adım Çözüm

1

Distribute the first term of the binomial,

3a

, to every term in the trinomial.

3a(4a^2) + 3a(5ab) + 3a(-b^2) = 12a^3 + 15a^2b - 3ab^2

This applies the distributive property to begin multiplying the polynomials.

2

Distribute the second term of the binomial,

-2b

, to every term in the trinomial.

-2b(4a^2) - 2b(5ab) - 2b(-b^2) = -8a^2b - 10ab^2 + 2b^3

This completes the distribution process for all terms.

3

Write out the full expanded expression by combining the results from the previous steps.

12a^3 + 15a^2b - 3ab^2 - 8a^2b - 10ab^2 + 2b^3

This provides the complete polynomial before simplification.

4

Group and combine the like terms.

12a^3 + (15a^2b - 8a^2b) + (-3ab^2 - 10ab^2) + 2b^3 = 12a^3 + 7a^2b - 13ab^2 + 2b^3

Combining terms with identical variable parts simplifies the expression to its final standard form.

Anahtar Kavram

Multiplying a binomial by a polynomial requires distributing every term in the first polynomial to every term in the second polynomial, carefully tracking negative signs, and then combining any like terms.

Tahmini Süre:1m 30s

Multiplying Polynomials

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