10 Questions
What is the result of multiplying $(4x^2 + 5x + 3) * (3x^2 - 2x + 1)$?
$12x^4 + 7x^3 - 5x^2 + 7x + 3$
In the expression $(2x^3 + x^2 - 5x) + (3x^3 - 2x^2 + 4)$, what is the sum of like terms?
$6x^3 - x^2 - x$
When subtracting $(7x^2 - 3x + 5) - (4x^2 + 2x - 1)$, what is the result?
$3x^2 - x + 6$
Which classification does a polynomial of the form $4x^4 - x^3 + 2$ fall into?
Quartic polynomial
What is the factored form of the quadratic expression $x^2 + 7x + 10$?
$(x + 6)(x + 4)$
If $f(x) = 3x^3 - x^2 + 7$, what is the degree of the polynomial function $f(x)$?
3
What is the sum of $(6x^2 - 4) + (7x^2 - x + 9)$?
$(13x^2 - x + 5)$
Subtract $(8x^3 - x^2 + x) - (4x^3 - x^2 + x)$ to get:
$4x^3$
What degree is a polynomial with the highest power term being $5x^{10}$?
10
Factorize $4x^2 - 12$ to get:
$(4)(x+3)(x-3)$
Study Notes
Polynomial Operations and Quadratic Functions
Polynomial operations and quadratic functions are essential concepts in algebra and are used to model various real-world scenarios. In this article, we will discuss the basics of polynomial operations, multiplying and adding polynomials, subtracting polynomials, classifying polynomials, factoring quadratic expressions, using the quadratic formula, and solving for zeros on quadratic functions.
Multiplying Polynomials
Multiplying polynomials involves multiplying each term in one polynomial by each term in the other polynomial, then adding the products. For example:
(2x^2 + 3x + 4) * (x^2 + 2x + 3) = 2x^4 + 5x^3 + 10x^2 + 6x + 12
Adding Polynomials
Adding polynomials involves adding the coefficients of like terms. For example:
(3x^2 + 2x + 4) + (5x^2 - 2x + 1) = 8x^2 + 3x - 3
Subtracting Polynomials
Subtracting polynomials involves subtracting the coefficients of like terms. For example:
(3x^2 + 2x + 4) - (5x^2 - 2x + 1) = 8x^2 - 3x + 5
Classifying Polynomials
Polynomials can be classified based on their degree, which is the highest power of the variable in the polynomial. For example, a polynomial of the form ax^2 + bx + c is a quadratic polynomial (degree 2).
Factoring Quadratic Expressions
Factoring quadratic expressions involves finding the factors of the polynomial. For example:
x^2 + 2x + 1 = (x+1)(x+1)
Using the Quadratic Formula
The quadratic formula is used to find the roots (or zeros) of a quadratic equation. The formula is:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
This formula can be used to solve equations like ax^2 + bx + c = 0.
Solving for Zeros on Quadratic Functions
Solving for zeros on quadratic functions involves finding the values of x that make the equation equal to zero. For example:
x^2 + 2x + 1 = 0
To find the roots of this equation, we can use the quadratic formula:
x = (-2 ± sqrt(2^2 - 4(1)(1))) / (2(1))
x = (-2 ± sqrt(4)) / 2
x = (-2 ± 2) / 2
x = -1 and x = 1
These are the zeros of the quadratic function x^2 + 2x + 1.
Learn about polynomial operations, including multiplying, adding, and subtracting polynomials, as well as classifying polynomials based on their degree. Explore how to factor quadratic expressions, utilize the quadratic formula to find roots, and solve for zeros on quadratic functions.
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