Answer :
Let's go through the questions step by step:
Question 5: Simplified form of 2 is 64.
It seems like there might be a typo or missing information here. Simplifying the number 2 to 64 doesn't make sense. If there's more context or a ratio/operation intended, it's not clear from this statement alone.
Question II: [tex]\( f(x) = 7 - 2x \)[/tex] and [tex]\( g(x) = x - 6 \)[/tex], then [tex]\( (f-g)(x) \)[/tex].
To find [tex]\( (f-g)(x) \)[/tex], you subtract [tex]\( g(x) \)[/tex] from [tex]\( f(x) \)[/tex].
1. [tex]\( f(x) = 7 - 2x \)[/tex]
2. [tex]\( g(x) = x - 6 \)[/tex]
3. Calculate [tex]\( (f-g)(x) = f(x) - g(x) = (7 - 2x) - (x - 6) \)[/tex].
4. Simplify the expression: [tex]\( (7 - 2x) - (x - 6) = 7 - 2x - x + 6 = 13 - 3x \)[/tex].
So, [tex]\( (f-g)(x) = 13 - 3x \)[/tex].
Question 6: The vertex of the graph of [tex]\( f(x) = -(x+1)^2 \)[/tex].
The function [tex]\( f(x) = -(x+1)^2 \)[/tex] is already in vertex form, which is [tex]\( f(x) = a(x-h)^2 + k \)[/tex].
1. Here, [tex]\( a = -1 \)[/tex], [tex]\( h = -1 \)[/tex], and [tex]\( k = 0 \)[/tex].
2. The vertex is at [tex]\( (h, k) \)[/tex], which is [tex]\((-1, 0)\)[/tex].
So, the vertex of the graph is [tex]\((-1, 0)\)[/tex].
Question 7: The leading coefficient of the polynomial function [tex]\( f(x) = 3x^4 + 2x^2 - 3 \)[/tex].
1. The leading term of [tex]\( f(x) = 3x^4 + 2x^2 - 3 \)[/tex] is [tex]\( 3x^4 \)[/tex].
2. The coefficient of the leading term (highest power of [tex]\( x \)[/tex]) is 3.
Thus, the leading coefficient is 3.
Question 8: Clarification needed on function [tex]\( f(x) = x^3 + 3x \)[/tex] and [tex]\( g(x) = x-x^2 \)[/tex].
Without clarity on what exactly is being asked, if operations between these functions are needed, please provide more information.
Question 10: Remainder of [tex]\( f(x) = 55x^{200} + 50 \)[/tex] when divided by [tex]\( d(x) = x + 1 \)[/tex].
Use the Remainder Theorem, which says the remainder of [tex]\( f(x) \)[/tex] when divided by [tex]\( x - c \)[/tex] is [tex]\( f(c) \)[/tex].
1. Here [tex]\( c = -1 \)[/tex] since [tex]\( d(x) = x + 1 \)[/tex] (equivalent to [tex]\( x - (-1) \)[/tex]).
2. Substitute [tex]\( x = -1 \)[/tex] into [tex]\( f(x) = 55(-1)^{200} + 50 = 55 \times 1 + 50 = 105 \)[/tex].
Therefore, the remainder is 105.
Question 11: If [tex]\( f(c) = 0 \)[/tex], then [tex]\( (x-c) \)[/tex] is a factor.
This statement is directly from the factor theorem. If [tex]\( f(c) = 0 \)[/tex], then [tex]\( x-c \)[/tex] is a factor.
Question 12: Multiplicity of polynomial [tex]\( g(x) = 5(x+\sqrt{2})^2(x + 2^3)(1+3x) \)[/tex].
Multiplicity refers to the number of times a factor appears in the polynomial:
1. [tex]\((x+\sqrt{2})^2\)[/tex] has a multiplicity of 2.
2. [tex]\( (x + 2^3) = (x + 8) \)[/tex] and [tex]\((1 + 3x)\)[/tex] both appear once, so they each have a multiplicity of 1.
The multiplicity of all factors are 2, 1, and 1, respectively.
Value of [tex]\( (3)^{-3} \)[/tex]:
Calculate the power:
1. [tex]\( (3)^{-3} = \frac{1}{3^3} \)[/tex].
2. [tex]\( 3^3 = 27 \)[/tex].
3. Therefore, [tex]\( \frac{1}{27} \)[/tex].
Question III: Choose the correct answers
Question 14: Linear function?
- A linear function has the form [tex]\(f(x) = ax + b\)[/tex].
- [tex]\(k(x) = 7\)[/tex] is a constant function, which is a special type of linear function with [tex]\(a = 0\)[/tex].
Question 15: Degree of [tex]\( f(x) = (x+3)(x^3+1) \)[/tex].
- Expansion gives: the highest power term is [tex]\(x^4\)[/tex], so degree 4.
Question 16: Which is a polynomial function?
- [tex]\(g(x) = 4x^{-2} + 3x - 7\)[/tex] is not a polynomial because [tex]\(x^{-2}\)[/tex] is negative.
- [tex]\(k(x) = 2^x\)[/tex] is exponential, not a polynomial.
- [tex]\(\sqrt{x}\)[/tex] is not polynomial due to the root.
- [tex]\(h(x)=3-x^6\)[/tex] is a polynomial function. Answer: D. [tex]\(h(x) = 3 - x^6\)[/tex].
Question 17: Coefficient of [tex]\( x^3 \)[/tex].
- Expression: [tex]\(\frac{7 - 12x^3 + 4x^4 - 2x^4 + 8}{4}\)[/tex].
- Combine like terms: [tex]\(-2x^4 + 4x^4 = 2x^4 \)[/tex], simplify expression
- Coefficient of [tex]\( x^3 \)[/tex] is [tex]\(-12/4 = -3\)[/tex]. Answer: B. [tex]\(-3\)[/tex].
Question 18: Value of [tex]\( f-g \)[/tex].
- Given [tex]\( f(x) = -2x^3 + 5x^2 + 3x + 2 \)[/tex]
- Given [tex]\( g(x) = -2x^3 + 4x^2 + 8x + 7 \)[/tex]
- Calculate [tex]\( f(x) - g(x) = (5x^2 + 3x + 2) - (4x^2 + 8x + 7) \)[/tex].
- Result: [tex]\( x^2 - 5x - 5 \)[/tex]. Answer: D.
Question 19: Remainder when [tex]\( x^2-x+3 \)[/tex] is divided by [tex]\(x - c\)[/tex], with [tex]\(c = -2\)[/tex].
- Remainder Theorem: Substitute [tex]\((-2)\)[/tex] into the polynomial.
- [tex]\( f(-2) = (-2)^2 - (-2) + 3 = 4 + 2 + 3 = 9 \)[/tex].
- The remainder is 9. Answer: C. 9.
Question 5: Simplified form of 2 is 64.
It seems like there might be a typo or missing information here. Simplifying the number 2 to 64 doesn't make sense. If there's more context or a ratio/operation intended, it's not clear from this statement alone.
Question II: [tex]\( f(x) = 7 - 2x \)[/tex] and [tex]\( g(x) = x - 6 \)[/tex], then [tex]\( (f-g)(x) \)[/tex].
To find [tex]\( (f-g)(x) \)[/tex], you subtract [tex]\( g(x) \)[/tex] from [tex]\( f(x) \)[/tex].
1. [tex]\( f(x) = 7 - 2x \)[/tex]
2. [tex]\( g(x) = x - 6 \)[/tex]
3. Calculate [tex]\( (f-g)(x) = f(x) - g(x) = (7 - 2x) - (x - 6) \)[/tex].
4. Simplify the expression: [tex]\( (7 - 2x) - (x - 6) = 7 - 2x - x + 6 = 13 - 3x \)[/tex].
So, [tex]\( (f-g)(x) = 13 - 3x \)[/tex].
Question 6: The vertex of the graph of [tex]\( f(x) = -(x+1)^2 \)[/tex].
The function [tex]\( f(x) = -(x+1)^2 \)[/tex] is already in vertex form, which is [tex]\( f(x) = a(x-h)^2 + k \)[/tex].
1. Here, [tex]\( a = -1 \)[/tex], [tex]\( h = -1 \)[/tex], and [tex]\( k = 0 \)[/tex].
2. The vertex is at [tex]\( (h, k) \)[/tex], which is [tex]\((-1, 0)\)[/tex].
So, the vertex of the graph is [tex]\((-1, 0)\)[/tex].
Question 7: The leading coefficient of the polynomial function [tex]\( f(x) = 3x^4 + 2x^2 - 3 \)[/tex].
1. The leading term of [tex]\( f(x) = 3x^4 + 2x^2 - 3 \)[/tex] is [tex]\( 3x^4 \)[/tex].
2. The coefficient of the leading term (highest power of [tex]\( x \)[/tex]) is 3.
Thus, the leading coefficient is 3.
Question 8: Clarification needed on function [tex]\( f(x) = x^3 + 3x \)[/tex] and [tex]\( g(x) = x-x^2 \)[/tex].
Without clarity on what exactly is being asked, if operations between these functions are needed, please provide more information.
Question 10: Remainder of [tex]\( f(x) = 55x^{200} + 50 \)[/tex] when divided by [tex]\( d(x) = x + 1 \)[/tex].
Use the Remainder Theorem, which says the remainder of [tex]\( f(x) \)[/tex] when divided by [tex]\( x - c \)[/tex] is [tex]\( f(c) \)[/tex].
1. Here [tex]\( c = -1 \)[/tex] since [tex]\( d(x) = x + 1 \)[/tex] (equivalent to [tex]\( x - (-1) \)[/tex]).
2. Substitute [tex]\( x = -1 \)[/tex] into [tex]\( f(x) = 55(-1)^{200} + 50 = 55 \times 1 + 50 = 105 \)[/tex].
Therefore, the remainder is 105.
Question 11: If [tex]\( f(c) = 0 \)[/tex], then [tex]\( (x-c) \)[/tex] is a factor.
This statement is directly from the factor theorem. If [tex]\( f(c) = 0 \)[/tex], then [tex]\( x-c \)[/tex] is a factor.
Question 12: Multiplicity of polynomial [tex]\( g(x) = 5(x+\sqrt{2})^2(x + 2^3)(1+3x) \)[/tex].
Multiplicity refers to the number of times a factor appears in the polynomial:
1. [tex]\((x+\sqrt{2})^2\)[/tex] has a multiplicity of 2.
2. [tex]\( (x + 2^3) = (x + 8) \)[/tex] and [tex]\((1 + 3x)\)[/tex] both appear once, so they each have a multiplicity of 1.
The multiplicity of all factors are 2, 1, and 1, respectively.
Value of [tex]\( (3)^{-3} \)[/tex]:
Calculate the power:
1. [tex]\( (3)^{-3} = \frac{1}{3^3} \)[/tex].
2. [tex]\( 3^3 = 27 \)[/tex].
3. Therefore, [tex]\( \frac{1}{27} \)[/tex].
Question III: Choose the correct answers
Question 14: Linear function?
- A linear function has the form [tex]\(f(x) = ax + b\)[/tex].
- [tex]\(k(x) = 7\)[/tex] is a constant function, which is a special type of linear function with [tex]\(a = 0\)[/tex].
Question 15: Degree of [tex]\( f(x) = (x+3)(x^3+1) \)[/tex].
- Expansion gives: the highest power term is [tex]\(x^4\)[/tex], so degree 4.
Question 16: Which is a polynomial function?
- [tex]\(g(x) = 4x^{-2} + 3x - 7\)[/tex] is not a polynomial because [tex]\(x^{-2}\)[/tex] is negative.
- [tex]\(k(x) = 2^x\)[/tex] is exponential, not a polynomial.
- [tex]\(\sqrt{x}\)[/tex] is not polynomial due to the root.
- [tex]\(h(x)=3-x^6\)[/tex] is a polynomial function. Answer: D. [tex]\(h(x) = 3 - x^6\)[/tex].
Question 17: Coefficient of [tex]\( x^3 \)[/tex].
- Expression: [tex]\(\frac{7 - 12x^3 + 4x^4 - 2x^4 + 8}{4}\)[/tex].
- Combine like terms: [tex]\(-2x^4 + 4x^4 = 2x^4 \)[/tex], simplify expression
- Coefficient of [tex]\( x^3 \)[/tex] is [tex]\(-12/4 = -3\)[/tex]. Answer: B. [tex]\(-3\)[/tex].
Question 18: Value of [tex]\( f-g \)[/tex].
- Given [tex]\( f(x) = -2x^3 + 5x^2 + 3x + 2 \)[/tex]
- Given [tex]\( g(x) = -2x^3 + 4x^2 + 8x + 7 \)[/tex]
- Calculate [tex]\( f(x) - g(x) = (5x^2 + 3x + 2) - (4x^2 + 8x + 7) \)[/tex].
- Result: [tex]\( x^2 - 5x - 5 \)[/tex]. Answer: D.
Question 19: Remainder when [tex]\( x^2-x+3 \)[/tex] is divided by [tex]\(x - c\)[/tex], with [tex]\(c = -2\)[/tex].
- Remainder Theorem: Substitute [tex]\((-2)\)[/tex] into the polynomial.
- [tex]\( f(-2) = (-2)^2 - (-2) + 3 = 4 + 2 + 3 = 9 \)[/tex].
- The remainder is 9. Answer: C. 9.
