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Questions and Answers
If a cubic polynomial has zeros of 2, -3, and 4, which of the following could be the polynomial?
If a cubic polynomial has zeros of 2, -3, and 4, which of the following could be the polynomial?
- $x^3 + 3x^2 + 10x + 24$
- $x^3 - 3x^2 - 10x + 24$ (correct)
- $x^3 - 3x^2 + 10x - 24$
- $x^3 + 3x^2 - 10x - 24$
A cubic polynomial has zeros of $\frac{1}{2}$, 1, and -3. Which of the following is a possible form of this polynomial?
A cubic polynomial has zeros of $\frac{1}{2}$, 1, and -3. Which of the following is a possible form of this polynomial?
- $2x^3 + 3x^2 - 5x + 3$
- $2x^3 - x^2 - 5x + 3$
- $2x^3 + x^2 - 5x - 3$ (correct)
- $2x^3 + x^2 + 5x - 3$
A cubic polynomial has a sum of zeros equal to 5, a sum of the product of its zeros taken two at a time equal to -2, and a product of its zeros equal to -24. Which polynomial satisfies these conditions?
A cubic polynomial has a sum of zeros equal to 5, a sum of the product of its zeros taken two at a time equal to -2, and a product of its zeros equal to -24. Which polynomial satisfies these conditions?
- $x^3 - 5x^2 + 2x - 24$
- $x^3 + 5x^2 - 2x - 24$
- $x^3 - 5x^2 - 2x + 24$ (correct)
- $x^3 + 5x^2 + 2x + 24$
When $f(x) = x^3 - 3x^2 + 5x - 3$ is divided by $g(x) = x^2 - 2$, what is the quotient?
When $f(x) = x^3 - 3x^2 + 5x - 3$ is divided by $g(x) = x^2 - 2$, what is the quotient?
When $f(x) = x^4 - 3x^2 + 4x + 5$ is divided by $g(x) = x^2 + 1 - x$, what is the remainder?
When $f(x) = x^4 - 3x^2 + 4x + 5$ is divided by $g(x) = x^2 + 1 - x$, what is the remainder?
When $f(x) = x^4 - 5x + 6$ is divided by $g(x) = 2 - x^2$, what is the quotient?
When $f(x) = x^4 - 5x + 6$ is divided by $g(x) = 2 - x^2$, what is the quotient?
Given that $x^2 - 3$ is a factor of $2x^4 + 3x^3 - 2x^2 - 9x - 12$, what is the result of the division?
Given that $x^2 - 3$ is a factor of $2x^4 + 3x^3 - 2x^2 - 9x - 12$, what is the result of the division?
If the polynomial $x^4 + 2x^3 + 8x^2 + 12x + 18$ is divided by $x^2 + 5$ and the remainder is of the form $px + q$, what are the values of $p$ and $q$?
If the polynomial $x^4 + 2x^3 + 8x^2 + 12x + 18$ is divided by $x^2 + 5$ and the remainder is of the form $px + q$, what are the values of $p$ and $q$?
When $3x^3 + x^2 + 2x + 5$ is divided by a polynomial $g(x)$, the quotient is $3x - 5$ and the remainder is $9x + 10$. What is the polynomial $g(x)$?
When $3x^3 + x^2 + 2x + 5$ is divided by a polynomial $g(x)$, the quotient is $3x - 5$ and the remainder is $9x + 10$. What is the polynomial $g(x)$?
Given $f(x) = 8 + 20x + x^2 - 6x^3$ and $g(x) = 2 + 5x - 3x^2$, which statement verifies the division algorithm?
Given $f(x) = 8 + 20x + x^2 - 6x^3$ and $g(x) = 2 + 5x - 3x^2$, which statement verifies the division algorithm?
If -1 is one of the zeros of the polynomial $x^3 + 2x^2 - 11x - 12$, what are the other zeros?
If -1 is one of the zeros of the polynomial $x^3 + 2x^2 - 11x - 12$, what are the other zeros?
If 1 and -2 are two zeros of the polynomial $x^3 - 4x^2 - 7x + 10$, what is the third zero?
If 1 and -2 are two zeros of the polynomial $x^3 - 4x^2 - 7x + 10$, what is the third zero?
If 3 and -3 are two zeros of the polynomial $x^4 + x^3 - 11x^2 - 9x + 18$, what are the other zeros?
If 3 and -3 are two zeros of the polynomial $x^4 + x^3 - 11x^2 - 9x + 18$, what are the other zeros?
If 2 and -2 are two zeros of the polynomial $2x^4 - 5x^3 - 11x^2 + 20x + 12$, what are the other zeros?
If 2 and -2 are two zeros of the polynomial $2x^4 - 5x^3 - 11x^2 + 20x + 12$, what are the other zeros?
For a cubic polynomial $p(x) = ax^3 + bx^2 + cx + d$ with zeros $\alpha$, $\beta$, and $\gamma$, which of the following relationships between the zeros and the coefficients is correct?
For a cubic polynomial $p(x) = ax^3 + bx^2 + cx + d$ with zeros $\alpha$, $\beta$, and $\gamma$, which of the following relationships between the zeros and the coefficients is correct?
Which of the following statements is true regarding polynomial division?
Which of the following statements is true regarding polynomial division?
Given a polynomial $f(x)$ and a factor $(x - a)$, what does the Factor Theorem state?
Given a polynomial $f(x)$ and a factor $(x - a)$, what does the Factor Theorem state?
If a polynomial $p(x)$ is divided by $(x - a)$ and the remainder is $R$, what does the Remainder Theorem state?
If a polynomial $p(x)$ is divided by $(x - a)$ and the remainder is $R$, what does the Remainder Theorem state?
Consider the polynomial division algorithm: Dividend = (Divisor × Quotient) + Remainder. Which of these must be true?
Consider the polynomial division algorithm: Dividend = (Divisor × Quotient) + Remainder. Which of these must be true?
Flashcards
What defines a cubic polynomial?
What defines a cubic polynomial?
A cubic polynomial has a degree of 3, meaning the highest power of the variable x is 3.
How to verify a zero of a polynomial?
How to verify a zero of a polynomial?
To verify if a number is a zero of a polynomial, substitute the number for the variable in the polynomial. If the result is zero, then the number is a zero of the polynomial.
What are coefficients?
What are coefficients?
Coefficients are the numerical or constant quantity placed before and multiplying the variable in an algebraic expression.
Relation between zeros and coefficients?
Relation between zeros and coefficients?
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What is polynomial division?
What is polynomial division?
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What is the division algorithm?
What is the division algorithm?
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What is the factor theorem?
What is the factor theorem?
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How to find all zeros of a polynomial?
How to find all zeros of a polynomial?
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Study Notes
- These notes cover exercises on polynomials
- There are questions to verify zeros and relationships, find cubic polynomials, perform division, and apply the division algorithm
Verifying Zeros and Relationships
- 3, -2, and 1 are zeros of the cubic polynomial p(x) = x³ - 2x² - 5x + 6
- It is important to verify the relationship between these zeros and the coefficients
- Similarly, 5, -2, and 1/3 are zeros of the polynomial p(x) = 3x³ - 10x² - 27x + 10
- The relationship between zeros and coefficients should also be verified for this polynomial
Finding Cubic Polynomials
- Need to determine a cubic polynomial given the zeros 2, -3, and 4
- Need to determine a cubic polynomial given the zeros 1/2, 1, and -3
- Determine a cubic polynomial given:
- The sum of its zeros
- The sum of the product of its zeros taken two at a time
- The product of its zeros are 5, -2, and -24 respectively
Polynomial Division
- Find the quotient and remainder when f(x) = x³ - 3x² + 5x - 3 is divided by g(x) = x² - 2
- Find the quotient and remainder when f(x) = x⁴ - 3x² + 4x + 5 is divided by g(x) = x² + 1 - x
- Find the quotient and remainder when f(x) = x⁴ - 5x + 6 is divided by g(x) = 2 - x²
- Show that x² - 3 is a factor of 2x⁴ + 3x³ - 2x² - 9x - 12 using polynomial division
- The polynomial (x⁴ + 2x³ + 8x² + 12x + 18) divided by (x² + 5) has a remainder of (px + q)
- Determine the values of p and q
Applying the Division Algorithm
- When 3x⁴ + x³ + 2x + 5 is divided by g(x), the quotient and remainder are (3x - 5) and (9x + 10) respectively
- Find g(x)
- Verify the division algorithm for f(x) = 8 + 20x + x² - 6x³ and g(x) = 2 + 5x - 3x²
Finding Zeros with Given Information
- Given that -1 is a zero of the polynomial x³ + 2x² - 11x - 12, find all the zeros
- Given 1 and -2 are zeros of x³ - 4x² - 7x + 10, find the third zero
- Given 3 and -3 are zeros of x⁴ + x³ - 11x² - 9x + 18, find all zeros
- Given 2 and -2 are zeros of 2x⁴ - 5x³ - 11x² + 20x + 12, find all zeros
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