Podcast
Questions and Answers
Which of the following is the correct complete factorization of $x^2(x+2) - 16(x+2)$?
Which of the following is the correct complete factorization of $x^2(x+2) - 16(x+2)$?
- $(x^2 + 16)(x+2)$
- $(x+4)(x-4)(x+2)$ (correct)
- $(x^2-16)(x+2)$
- $(x+4)(x-4)$
What are the zeroes of the polynomial $(x+4)(x-4)(5x+6)(5x-6) = 0$?
What are the zeroes of the polynomial $(x+4)(x-4)(5x+6)(5x-6) = 0$?
- $x = -4, 4, -5/6, 5/6$
- $x = -4, 4, -5, 5$
- $x = -4, 4, -6/5, 6/5$ (correct)
- $x = -4, -4, -6, 6$
Given the vertex form of a quadratic equation $y = a(x - 3)^2 + 5$, and the point (0, -4) on the parabola, what is the value of 'a'?
Given the vertex form of a quadratic equation $y = a(x - 3)^2 + 5$, and the point (0, -4) on the parabola, what is the value of 'a'?
- -2
- 2
- 1
- -1 (correct)
For the quadratic equation derived from vertex form $0 = -(x - 3)^2 + 5$, what are the real solutions for x?
For the quadratic equation derived from vertex form $0 = -(x - 3)^2 + 5$, what are the real solutions for x?
Given the polynomial $f(x) = x^4 - 2x^3 + 28x^2 - 72x - 288$, and knowing x = 4 and x = -2 are real roots, what are the remaining complex roots?
Given the polynomial $f(x) = x^4 - 2x^3 + 28x^2 - 72x - 288$, and knowing x = 4 and x = -2 are real roots, what are the remaining complex roots?
Given 3x^2 - 15x + 2x - 10, which of the following represents the correctly factored form?
Given 3x^2 - 15x + 2x - 10, which of the following represents the correctly factored form?
If a polynomial has factors of $(x+4)$, $(x-4)$, $(5x+6)$ and $(5x-6)$, what is the nature of its roots?
If a polynomial has factors of $(x+4)$, $(x-4)$, $(5x+6)$ and $(5x-6)$, what is the nature of its roots?
Given the equation $y = a(x-3)^2 + 5$, how does changing the value of 'a' affect the parabola's graph?
Given the equation $y = a(x-3)^2 + 5$, how does changing the value of 'a' affect the parabola's graph?
If synthetic division is used on the polynomial $f(x) = x^4 - 2x^3 + 28x^2 - 72x - 288$ with root x = 4, what is the resulting polynomial?
If synthetic division is used on the polynomial $f(x) = x^4 - 2x^3 + 28x^2 - 72x - 288$ with root x = 4, what is the resulting polynomial?
The complex roots of a polynomial are found to be $x = \pm 6i$. What does this imply about the graph of the polynomial function in the real coordinate plane?
The complex roots of a polynomial are found to be $x = \pm 6i$. What does this imply about the graph of the polynomial function in the real coordinate plane?
Flashcards
Factor by Grouping
Factor by Grouping
Grouping terms with common factors to simplify expressions.
Synthetic Division
Synthetic Division
Replacing a polynomial with a simpler equivalent expression.
Zeros of a Function
Zeros of a Function
Solutions to a polynomial equation where the function equals zero.
Complex Numbers
Complex Numbers
Signup and view all the flashcards
Study Notes
- Test 3, Version 2 is set for February 2025.
- The test has a total score of 43 points.
Factoring Completely
- Example a: x²(x+2) - 16(x+2) factors to (x²-16)(x+2), which further factors to (x+4)(x-4)(x+2).
- Example b: 3x² - 15x + 2x - 10 factors to 3x(x-5) + 2(x-5), which results in (3x+2)(x-5).
Division and Factoring
- (x+4)(x-4)(5x+6)(5x-6) = 0 gives the solutions x = 4, x = -4, x = -6/5, and x = 6/5.
Solving for y
- Given y = a(x-3)² + 5, using the point (0, -4) to solve for a yields a = -1.
- Therefore, y = -(x-3)² + 5. Setting y to 0 gives 0 = -(x-3)² + 5.
- Manipulating the equation -5 = -(x-3)² leads to 5 = (x-3)².
- Taking the square root gives ±√5 = x - 3, so x = 3 ± √5.
Finding Complex Zeros
- The function f(x) = x⁴ - 2x³ + 28x² - 72x - 288 has complex zeros.
- Using a calculator and graph, two real roots are found: x = -2 and x = 4.
- Applying synthetic division with x = 4 and x = -2 helps factor the polynomial.
- The synthetic division results in the quadratic x² + 36 = 0.
- Solving x² + 36 = 0 yields x² = -36, so x = ±6i.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Review for Test 3 covering factoring techniques and solving algebraic equations. Includes factoring completely, division and factoring, solving for y, and finding complex zeros. Examples provided for each method.
