Multiply the polynomial expressions (2x2−3)(x+5)

Question

.(1 point)
Responses

2x2−3x−15
2 x squared minus 3 x minus 15

2x3+10x2−3x−15
2 x cubed plus 10 x squared minus 3 x minus 15

3x2+7x−15
3 x squared plus 7 x minus 15

3x3+10x2−3x−15

Answer 1

To multiply the polynomial expressions \( (2x^2 - 3)(x + 5) \), we can use the distributive property (also known as the FOIL method for binomials).

1. Distribute \( 2x^2 \) to both terms in the second polynomial: \[ 2x^2 \cdot x + 2x^2 \cdot 5 = 2x^3 + 10x^2 \]

2. Distribute \(-3\) to both terms in the second polynomial: \[ -3 \cdot x + (-3) \cdot 5 = -3x - 15 \]

3. Combine all the results: \[ 2x^3 + 10x^2 - 3x - 15 \]

So, the final result of the multiplication \( (2x^2 - 3)(x + 5) \) is:

\[ 2x^3 + 10x^2 - 3x - 15 \]

The correct response is: 2x³ + 10x² - 3x - 15

Answer 2

Which of the following responses shows that polynomials form a closed system under multiplication?(1 point)
Responses

5x⋅2
5 x times 2

(13)(x4−2)
left parenthesis Start Fraction 1 over 3 End Fraction right parenthesis left parenthesis Start Fraction x over 4 End Fraction minus 2 right parenthesis

2x3+x2
2 x cubed plus x squared

5⋅3

Answer 3

To show that polynomials form a closed system under multiplication, we need an example where multiplying two polynomials results in another polynomial.

Looking at the provided responses:

1. 5x⋅2: This is the multiplication of a polynomial \(5x\) (which is a polynomial of degree 1) and a constant \(2\). The result will be \(10x\), which is still a polynomial.

2. (13)(x⁴−2): This involves multiplying a constant \(13\) by the polynomial \(x^4 - 2\). The result will be \(13x^4 - 26\), which is still a polynomial.

3. 2x³ + x²: This is a polynomial itself, but does not show multiplication.

4. 5⋅3: This is just the multiplication of two constants and does not involve polynomials.

Among these options, both 5x⋅2 and (13)(x⁴−2) show multiplication that results in another polynomial. However, the response that explicitly shows a polynomial being multiplied by another (the second option) is the most illustrative of the concept that polynomials are closed under multiplication. Thus, the best response that demonstrates that polynomials form a closed system under multiplication is:

(13)(x⁴−2).

Answer 4

What must be true about this multiplication problem?

(9x^3+7/6)(x−2/5)

(1 point)
Responses

The result is a polynomial.
The result is a polynomial.

The result is not a closed system.
The result is not a closed system.

The result is not a polynomial.
The result is not a polynomial.

The result is unknown.

Answer 5

To determine what must be true about the multiplication problem

\[ (9x^3 + \frac{7}{6})(x - \frac{2}{5}), \]

let's analyze the components:

1. First expression: \(9x^3 + \frac{7}{6}\) is a polynomial because it is a sum of terms, including a term \(9x^3\) (a degree 3 polynomial) and a constant term \(\frac{7}{6}\).

2. Second expression: \(x - \frac{2}{5}\) is also a polynomial (specifically, a degree 1 polynomial) because it consists of a variable term and a constant.

3. Multiplying Polynomials: When you multiply two polynomials, the result is always a polynomial. The degree of the resulting polynomial will be the sum of the degrees of the individual polynomials.

Therefore, the multiplication of these two polynomials will yield another polynomial.

Given these points, the correct response is:

The result is a polynomial.

Answer 6

Which of the following is the simplified form of 5x^2−40x+10/5
?(1 point)
Responses

x2−8x+2
x squared minus 8 x plus 2

5x−8
5 x minus 8

x3−8x2+2x
x cubed minus 8 x squared plus 2 x

x2−35x+5

Answer 7

To simplify the expression \( \frac{5x^2 - 40x + 10}{5} \), we will divide each term in the numerator by 5.

1. Start with the original expression: \[ \frac{5x^2 - 40x + 10}{5} \]

2. Divide each term in the numerator by 5: \[ \frac{5x^2}{5} - \frac{40x}{5} + \frac{10}{5} = x^2 - 8x + 2 \]

So, the simplified form of \( \frac{5x^2 - 40x + 10}{5} \) is:

\[ x^2 - 8x + 2 \]

Thus, the correct response is:

x² - 8x + 2.

Answer 8

Which of the following correctly divides the expressions −21x^3+14x^2−14x/7x
?(1 point)
Responses

−3x2+2x+2
negative 3 x squared plus 2 x plus 2

−3x2+2x−2
negative 3 x squared plus 2 x minus 2

3x2+2x−2
3 x squared plus 2 x minus 2

−2x2+2x−2

Answer 9

To divide the expression \(-21x^3 + 14x^2 - 14x\) by \(7x\), we can individually divide each term in the polynomial by \(7x\).

1. Start with the original expression: \[ \frac{-21x^3 + 14x^2 - 14x}{7x} \]

2. Divide each term:

   • The first term: \[ \frac{-21x^3}{7x} = -3x^2 \]
   • The second term: \[ \frac{14x^2}{7x} = 2x \]
   • The third term: \[ \frac{-14x}{7x} = -2 \]

3. Combine the results: \[ -3x^2 + 2x - 2 \]

Therefore, the correct response is:

−3x² + 2x − 2.

Answer 10

What is the quotient of the following?

28x^7+40x^6−16x^4/4x^3


(1 point)
( )x^( ) + ( ) ^( ) - ( )x

Answer 11

To find the quotient of the expression \(\frac{28x^7 + 40x^6 - 16x^4}{4x^3}\), we will divide each term in the numerator by \(4x^3\).

1. Divide each term:

   • For the first term: \[ \frac{28x^7}{4x^3} = \frac{28}{4}x^{7-3} = 7x^4 \]
   • For the second term: \[ \frac{40x^6}{4x^3} = \frac{40}{4}x^{6-3} = 10x^3 \]
   • For the third term: \[ \frac{-16x^4}{4x^3} = \frac{-16}{4}x^{4-3} = -4x \]

2. Combine the results: \[ 7x^4 + 10x^3 - 4x \]

Thus, the quotient can be written in the form:

\[ (7)x^{4} + (10)x^{3} - (4)x \]

So, filling in the blanks:

(7)x^(4) + (10)^(3) - (4)x.