Answered

Question 1:

An electric device delivers a current of [tex]15.0 \, \text{A}[/tex] for 30 seconds. How many electrons flow through it?

---

Question 2:

Which best explains why Irving sets "The Adventure of the Mysterious Stranger" in a land of "masks and gondolas"?

A. The setting is symbolic of the idea that a life of quiet study is the ideal pursuit.
B. The setting is symbolic of the idea that innocence cannot be outgrown.
C. The setting is symbolic of the idea that ease and affluence are available to all.
D. The setting is symbolic of the idea that appearances can be deceiving.

---

Question 3:

Read the lines from "The Tide Rises, The Tide Falls."

"Darkness settles on roofs and walls,
But the sea, the sea in darkness calls;"

The imagery in these lines evokes a sense of:

A. laziness
B. fear
C. mystery
D. despair

---

Question 4:

Solve for [tex]x[/tex].

[tex]\[ 3x = 6x - 2 \][/tex]

---

Question 5:

Two polynomials [tex]P(x)[/tex] and [tex]D(x)[/tex] are given. Use either synthetic division or long division to divide [tex]P(x)[/tex] by [tex]D(x)[/tex], and express [tex]P(x)[/tex] in the form [tex]P(x) = D(x) \cdot Q(x) + R(x)[/tex].

[tex]\[ P(x) = x^4 + 16x^3 + 9x^2, \quad D(x) = 4x^2 + 1 \][/tex]

---

Question 6:

If [tex]c[/tex] is a real zero of the polynomial [tex]P(x)[/tex], then all the other zeros of [tex]P(x)[/tex] are zeros of [tex]\frac{P(x)}{x - c}[/tex].

A. True
B. False



Answer :

GinnyAnswer
To divide the polynomial [tex]\(P(x)\)[/tex] by the polynomial [tex]\(D(x)\)[/tex] and express [tex]\(P(x)\)[/tex] in the form of [tex]\( P(x) = D(x) \cdot Q(x) + R(x) \)[/tex], we need to perform polynomial division.

### Given Polynomials
- [tex]\( P(x) = x^3 + 16x^2 + 9x \)[/tex]
- [tex]\( D(x) = 4x^2 + 1 \)[/tex]

### Polynomial Division
Step-by-Step Solution:

1. Set Up the Division:

Divide [tex]\( P(x) = x^3 + 16x^2 + 9x \)[/tex] by [tex]\( D(x) = 4x^2 + 1 \)[/tex].

2. Determine the First Term of the Quotient:

The first term in the quotient [tex]\(Q(x)\)[/tex] is determined by dividing the leading term of the dividend [tex]\(x^3\)[/tex] by the leading term of the divisor [tex]\(4x^2\)[/tex].

[tex]\[ \frac{x^3}{4x^2} = \frac{1}{4}x = 0.25x \][/tex]

3. Multiply and Subtract:

Multiply [tex]\(0.25x \cdot (4x^2 + 1)\)[/tex] and subtract from [tex]\(P(x)\)[/tex]:

[tex]\[ 0.25x \cdot (4x^2 + 1) = x^3 + 0.25x \][/tex]

Subtract this from [tex]\(P(x)\)[/tex]:

[tex]\[ (x^3 + 16x^2 + 9x) - (x^3 + 0.25x) = 16x^2 + 8.75x \][/tex]

4. Repeat the Process:

Next, divide [tex]\(16x^2\)[/tex] by [tex]\(4x^2\)[/tex] to get the next term of the quotient:

[tex]\[ \frac{16x^2}{4x^2} = 4 \][/tex]

Multiply and subtract:

[tex]\[ 4 \cdot (4x^2 + 1) = 16x^2 + 4 \][/tex]

Subtract from the remaining polynomial:

[tex]\[ (16x^2 + 8.75x) - (16x^2 + 4) = 8.75x - 4 \][/tex]

5. Final Quotient and Remainder:

The quotient [tex]\(Q(x)\)[/tex] is [tex]\(0.25x + 4\)[/tex], and the remainder [tex]\(R(x)\)[/tex] is [tex]\(8.75x - 4\)[/tex].

### Express [tex]\(P(x)\)[/tex] in the Required Form:

Using the results of our division, express [tex]\(P(x)\)[/tex]:

[tex]\[ P(x) = D(x) \cdot Q(x) + R(x) \][/tex]

Substitute the quotient and remainder:

[tex]\[ x^3 + 16x^2 + 9x = (4x^2 + 1)(0.25x + 4) + (8.75x - 4) \][/tex]

Therefore, [tex]\(P(x) = (4x^2 + 1)(0.25x + 4) + 8.75x - 4\)[/tex].

Other Questions