Answer :
Sure! Let's break down each part of the question and explore the solutions step-by-step.
### Question 10:
Find the remainder of [tex]\( f(x) = 55x^{200} + 50 \)[/tex] when divided by [tex]\( d(x) = x + 1 \)[/tex].
To find the remainder of a polynomial [tex]\( f(x) \)[/tex] when divided by [tex]\( x - a \)[/tex], we use the Remainder Theorem. According to this theorem, the remainder is [tex]\( f(a) \)[/tex].
For [tex]\( f(x) = 55x^{200} + 50 \)[/tex] and [tex]\( d(x) = x + 1 \)[/tex], the remainder can be found by evaluating [tex]\( f(-1) \)[/tex]:
[tex]\[
f(-1) = 55(-1)^{200} + 50 = 55 \times 1 + 50 = 105
\][/tex]
Therefore, the remainder is 50.
### Question 11:
If [tex]\( f(c) = 0 \)[/tex], then [tex]\( (x-c) \)[/tex] is a factor of [tex]\( f(x) \)[/tex].
This statement is based on the Factor Theorem, which states that if a polynomial [tex]\( f(x) \)[/tex] has a root [tex]\( c \)[/tex], then [tex]\( (x-c) \)[/tex] is a factor of [tex]\( f(x) \)[/tex].
### Question 12:
Find the multiplicity of [tex]\( g(x) = 5(x+\sqrt{2})^2(x+2^3)(1+3x) \)[/tex].
To find the multiplicity of the polynomial:
1. The term [tex]\( (x+\sqrt{2})^2 \)[/tex] indicates a root [tex]\( x = -\sqrt{2} \)[/tex] with multiplicity 2.
2. The term [tex]\( (x+2^3) \)[/tex] indicates a root [tex]\( x = -8 \)[/tex] with multiplicity 1.
3. The term [tex]\( (1+3x) \)[/tex] can be rewritten as [tex]\( (x + \frac{1}{3}) \)[/tex], indicating a root [tex]\( x = -\frac{1}{3} \)[/tex] with multiplicity 1.
Overall, the multiplicities are determined by these terms. Each term contributes to the overall polynomial's multiplicity.
### Question 3:
Find the value of [tex]\( \left(\frac{2}{3}\right)^{-3} \)[/tex].
To solve [tex]\( \left(\frac{2}{3}\right)^{-3} \)[/tex], we use the property of exponents that states [tex]\( a^{-n} = \frac{1}{a^n} \)[/tex]. Thus:
[tex]\[
\left(\frac{2}{3}\right)^{-3} = \left(\frac{3}{2}\right)^3 = \frac{3^3}{2^3} = \frac{27}{8} = 3.375
\][/tex]
### Question 4:
Determine which of the following is a linear function:
- A. [tex]\( f(x) = |x| \)[/tex] - This is not linear; it's a piecewise function.
- B. [tex]\( k(x) = 7 \)[/tex] - This is a linear function, specifically a constant function.
- C. [tex]\( l(x) = \sqrt{5}x + \sqrt{2} \)[/tex] - This is a linear function.
- D. [tex]\( h(x) = (x-1)(x+1) \)[/tex] - This is not linear; it is quadratic.
Thus, both [tex]\( B \)[/tex] and [tex]\( C \)[/tex] are linear functions.
### Question 5:
Determine the degree of [tex]\( f(x) = (x+3)(x^3+1) \)[/tex].
To find the degree of the polynomial, multiply the degrees of the factors:
- [tex]\( (x+3) \)[/tex] is degree 1
- [tex]\( (x^3+1) \)[/tex] is degree 3
Multiplying them gives:
[tex]\[
1 + 3 = 4
\][/tex]
Thus, the polynomial has a degree of 4.
### Question 6:
Which of the following is a polynomial function?
- A. [tex]\( g(x) = 4x^{-2} + 3x - 7 \)[/tex] - Not a polynomial; negative exponent.
- B. [tex]\( k(x) = 2^x \)[/tex] - Not a polynomial; exponential function.
- C. [tex]\( \sqrt{x} \)[/tex] - Not a polynomial; non-integer exponent.
- D. [tex]\( 3-x^5 \)[/tex] - This is a polynomial function.
Thus, the correct option is [tex]\( D \)[/tex].
### Question 7:
What is the coefficient of [tex]\( x^3 \)[/tex] in the expression [tex]\(\frac{7-12x^3+4x^4-2x^4+8}{4}\)[/tex]?
Simplify the expression first:
[tex]\[
7 - 12x^3 + 4x^4 - 2x^4 + 8 = 7 - 12x^3 + (4 - 2)x^4 + 8 = 7 - 12x^3 + 2x^4 + 8
\][/tex]
Divide by 4:
[tex]\[
\frac{7 + 8}{4} - \frac{12}{4}x^3 + \frac{2}{4}x^4 = \frac{15}{4} - 3x^3 + \frac{1}{2}x^4
\][/tex]
The coefficient of [tex]\( x^3 \)[/tex] is [tex]\(-3\)[/tex].
### Question 8:
Find the value of [tex]\( f(x) - g(x) \)[/tex] where [tex]\( f(x) = -2x^3 + 5x^2 + 3x + 2 \)[/tex] and [tex]\( g(x) = -2x^3 + 4x^2 + 8x + 7 \)[/tex].
Subtract [tex]\( g(x) \)[/tex] from [tex]\( f(x) \)[/tex]:
[tex]\[
f(x) - g(x) = \left(-2x^3 + 5x^2 + 3x + 2\right) - \left(-2x^3 + 4x^2 + 8x + 7\right)
\][/tex]
Simplifying:
[tex]\[
= -2x^3 + 5x^2 + 3x + 2 + 2x^3 - 4x^2 - 8x - 7
\][/tex]
Combine like terms:
[tex]\[
= 0x^3 + (5x^2 - 4x^2) + (3x - 8x) + (2 - 7)
\][/tex]
[tex]\[
= x^2 - 5x - 5
\][/tex]
### Question 9:
Determine the remainder when [tex]\( x^2 - x + 3 \)[/tex] is divided by [tex]\( x-c \)[/tex] with [tex]\( c = -2 \)[/tex].
By substituting [tex]\( c = -2 \)[/tex] into the expression:
[tex]\[
f(-2) = (-2)^2 - (-2) + 3 = 4 + 2 + 3 = 9
\][/tex]
Hence, the remainder is 9.
These are your answers with provided step-by-step explanations. Let me know if you have any more questions!
### Question 10:
Find the remainder of [tex]\( f(x) = 55x^{200} + 50 \)[/tex] when divided by [tex]\( d(x) = x + 1 \)[/tex].
To find the remainder of a polynomial [tex]\( f(x) \)[/tex] when divided by [tex]\( x - a \)[/tex], we use the Remainder Theorem. According to this theorem, the remainder is [tex]\( f(a) \)[/tex].
For [tex]\( f(x) = 55x^{200} + 50 \)[/tex] and [tex]\( d(x) = x + 1 \)[/tex], the remainder can be found by evaluating [tex]\( f(-1) \)[/tex]:
[tex]\[
f(-1) = 55(-1)^{200} + 50 = 55 \times 1 + 50 = 105
\][/tex]
Therefore, the remainder is 50.
### Question 11:
If [tex]\( f(c) = 0 \)[/tex], then [tex]\( (x-c) \)[/tex] is a factor of [tex]\( f(x) \)[/tex].
This statement is based on the Factor Theorem, which states that if a polynomial [tex]\( f(x) \)[/tex] has a root [tex]\( c \)[/tex], then [tex]\( (x-c) \)[/tex] is a factor of [tex]\( f(x) \)[/tex].
### Question 12:
Find the multiplicity of [tex]\( g(x) = 5(x+\sqrt{2})^2(x+2^3)(1+3x) \)[/tex].
To find the multiplicity of the polynomial:
1. The term [tex]\( (x+\sqrt{2})^2 \)[/tex] indicates a root [tex]\( x = -\sqrt{2} \)[/tex] with multiplicity 2.
2. The term [tex]\( (x+2^3) \)[/tex] indicates a root [tex]\( x = -8 \)[/tex] with multiplicity 1.
3. The term [tex]\( (1+3x) \)[/tex] can be rewritten as [tex]\( (x + \frac{1}{3}) \)[/tex], indicating a root [tex]\( x = -\frac{1}{3} \)[/tex] with multiplicity 1.
Overall, the multiplicities are determined by these terms. Each term contributes to the overall polynomial's multiplicity.
### Question 3:
Find the value of [tex]\( \left(\frac{2}{3}\right)^{-3} \)[/tex].
To solve [tex]\( \left(\frac{2}{3}\right)^{-3} \)[/tex], we use the property of exponents that states [tex]\( a^{-n} = \frac{1}{a^n} \)[/tex]. Thus:
[tex]\[
\left(\frac{2}{3}\right)^{-3} = \left(\frac{3}{2}\right)^3 = \frac{3^3}{2^3} = \frac{27}{8} = 3.375
\][/tex]
### Question 4:
Determine which of the following is a linear function:
- A. [tex]\( f(x) = |x| \)[/tex] - This is not linear; it's a piecewise function.
- B. [tex]\( k(x) = 7 \)[/tex] - This is a linear function, specifically a constant function.
- C. [tex]\( l(x) = \sqrt{5}x + \sqrt{2} \)[/tex] - This is a linear function.
- D. [tex]\( h(x) = (x-1)(x+1) \)[/tex] - This is not linear; it is quadratic.
Thus, both [tex]\( B \)[/tex] and [tex]\( C \)[/tex] are linear functions.
### Question 5:
Determine the degree of [tex]\( f(x) = (x+3)(x^3+1) \)[/tex].
To find the degree of the polynomial, multiply the degrees of the factors:
- [tex]\( (x+3) \)[/tex] is degree 1
- [tex]\( (x^3+1) \)[/tex] is degree 3
Multiplying them gives:
[tex]\[
1 + 3 = 4
\][/tex]
Thus, the polynomial has a degree of 4.
### Question 6:
Which of the following is a polynomial function?
- A. [tex]\( g(x) = 4x^{-2} + 3x - 7 \)[/tex] - Not a polynomial; negative exponent.
- B. [tex]\( k(x) = 2^x \)[/tex] - Not a polynomial; exponential function.
- C. [tex]\( \sqrt{x} \)[/tex] - Not a polynomial; non-integer exponent.
- D. [tex]\( 3-x^5 \)[/tex] - This is a polynomial function.
Thus, the correct option is [tex]\( D \)[/tex].
### Question 7:
What is the coefficient of [tex]\( x^3 \)[/tex] in the expression [tex]\(\frac{7-12x^3+4x^4-2x^4+8}{4}\)[/tex]?
Simplify the expression first:
[tex]\[
7 - 12x^3 + 4x^4 - 2x^4 + 8 = 7 - 12x^3 + (4 - 2)x^4 + 8 = 7 - 12x^3 + 2x^4 + 8
\][/tex]
Divide by 4:
[tex]\[
\frac{7 + 8}{4} - \frac{12}{4}x^3 + \frac{2}{4}x^4 = \frac{15}{4} - 3x^3 + \frac{1}{2}x^4
\][/tex]
The coefficient of [tex]\( x^3 \)[/tex] is [tex]\(-3\)[/tex].
### Question 8:
Find the value of [tex]\( f(x) - g(x) \)[/tex] where [tex]\( f(x) = -2x^3 + 5x^2 + 3x + 2 \)[/tex] and [tex]\( g(x) = -2x^3 + 4x^2 + 8x + 7 \)[/tex].
Subtract [tex]\( g(x) \)[/tex] from [tex]\( f(x) \)[/tex]:
[tex]\[
f(x) - g(x) = \left(-2x^3 + 5x^2 + 3x + 2\right) - \left(-2x^3 + 4x^2 + 8x + 7\right)
\][/tex]
Simplifying:
[tex]\[
= -2x^3 + 5x^2 + 3x + 2 + 2x^3 - 4x^2 - 8x - 7
\][/tex]
Combine like terms:
[tex]\[
= 0x^3 + (5x^2 - 4x^2) + (3x - 8x) + (2 - 7)
\][/tex]
[tex]\[
= x^2 - 5x - 5
\][/tex]
### Question 9:
Determine the remainder when [tex]\( x^2 - x + 3 \)[/tex] is divided by [tex]\( x-c \)[/tex] with [tex]\( c = -2 \)[/tex].
By substituting [tex]\( c = -2 \)[/tex] into the expression:
[tex]\[
f(-2) = (-2)^2 - (-2) + 3 = 4 + 2 + 3 = 9
\][/tex]
Hence, the remainder is 9.
These are your answers with provided step-by-step explanations. Let me know if you have any more questions!
