Perform the indicated operation. ( x 3 + 3 x 2 − 7 x ) + 2 ( x 3 − 5 x 2 + 9 x + 4 ) \left(x^3+3x^2-7x\right)+2\left(x^3-5x^2+9x+4\right) ( x 3 + 3 x 2 − 7 x ) + 2 ( x 3 − 5 x 2 + 9 x + 4 )

3 x 3 − 7 x 2 + 11 x + 8 3x^3-7x^2+11x+8 3 x 3 − 7 x 2 + 11 x + 8

2 x 3 − 2 x 2 + 2 x + 6 2x^3-2x^2+2x+6 2 x 3 − 2 x 2 + 2 x + 6

2 x 3 − 2 x 2 + 2 x + 4 2x^3-2x^2+2x+4 2 x 3 − 2 x 2 + 2 x + 4

3 x 3 − 2 x 2 + 2 x + 4 3x^3-2x^2+2x+4 3 x 3 − 2 x 2 + 2 x + 4

Perform the indicated operation. ( − 2 x 4 + 10 x 3 + 6 x − 3 ) − ( x 4 − 7 x 2 + 8 x + 5 ) \left(-2x^4+10x^3+6x-3\right)-\left(x^4-7x^2+8x+5\right) ( − 2 x 4 + 10 x 3 + 6 x − 3 ) − ( x 4 − 7 x 2 + 8 x + 5 )

− 3 x 4 + 10 x 3 + 7 x 2 − 2 x − 8 -3x^4+10x^3+7x^2-2x-8 − 3 x 4 + 10 x 3 + 7 x 2 − 2 x − 8

− 3 x 4 + 17 x 3 − 2 x − 8 -3x^4+17x^3-2x-8 − 3 x 4 + 17 x 3 − 2 x − 8

− 3 x 4 + 17 x 2 − 2 x − 8 -3x^4+17x^2-2x-8 − 3 x 4 + 17 x 2 − 2 x − 8

− x 4 + 10 x 3 − 7 x 2 + 14 x + 2 -x^4+10x^3-7x^2+14x+2 − x 4 + 10 x 3 − 7 x 2 + 14 x + 2

0. Review of Algebra

Polynomials Intro

0. Review of Algebra

Polynomials Intro: Videos & Practice Problems

Video Lessons Practice Worksheet

Topic summary

Polynomials are algebraic expressions with variables raised to positive whole number exponents. They can be classified as monomials, binomials, or trinomials based on the number of terms. To express polynomials in standard form, arrange terms in descending order of their degree, which is determined by the highest exponent. When adding or subtracting polynomials, combine like terms carefully, especially when distributing negative signs. Understanding coefficients, constants, and the structure of polynomials is essential for further operations and simplifications.

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Introduction to Polynomials

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Introduction to Polynomials Video Summary

Polynomials are a specific type of algebraic expression characterized by having variables raised only to positive whole number exponents. For instance, expressions like $6x^3$, $3x^2$, and $5x$ are all polynomials because their exponents (3, 2, and 1, respectively) are positive whole numbers. If an expression contains a negative or fractional exponent, such as $2x^{-3}$, it does not qualify as a polynomial.

Polynomials can be categorized based on the number of terms they contain. A polynomial with one term is called a monomial, while one with two terms is referred to as a binomial, and one with three terms is known as a trinomial. The prefixes "mono-", "bi-", and "tri-" indicate the number of terms, similar to how they are used in words like "monorail," "bicycle," and "tricycle."

To determine if an expression is a polynomial, examine the exponents of its variables. For example, the expression $ \frac{3}{4}x + x^3 $ is a polynomial because both terms have positive whole number exponents. It consists of two terms, making it a binomial. Conversely, the expression $ \frac{5}{y} $ is not a polynomial because it can be rewritten as $5y^{-1}$, which includes a negative exponent. Lastly, the expression $2x^3y^2$ is a polynomial with two variables, and since it has no plus or minus signs separating terms, it is classified as a monomial.

Understanding these definitions and classifications is crucial as they form the foundation for further operations with polynomials, such as addition, subtraction, and multiplication, which will be explored later in the course.

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Standard Form of Polynomials

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Standard Form of Polynomials Video Summary

Understanding polynomials is essential for simplifying expressions and writing them in standard form. Standard form requires that all terms of a polynomial are arranged in descending order based on their exponents. For example, in the polynomial 3x² + 5x + 4, the terms are organized from the highest exponent (2) to the lowest (0), with the constant term 4 representing x⁰.

The degree of a polynomial is defined as the highest exponent of the variable present. In the example above, the degree is 2, indicating it is a second-degree polynomial. Additionally, it is important to recognize the roles of coefficients and constants: coefficients are the numbers multiplying the variables, while constants are standalone numbers. The leading coefficient is the coefficient of the term with the highest exponent, which in this case is 3.

To convert a polynomial into standard form, one must ensure that all like terms are combined and that the terms are arranged in the correct order. For instance, consider the expression 1/2x + x³. To write this in standard form, rearranging it gives x³ + 1/2x, where the degree is 3 and the leading coefficient is 1.

In more complex expressions, such as -3x² + x² + 5x + 2x - 7, it is crucial to maintain the signs of each term while rearranging. After organizing the terms in descending order, the expression becomes -3x² + 1x² + 7x - 7. Combining like terms results in -2x² + 7x - 7, which is now in standard form. Here, the degree is 2, and the leading coefficient is -2.

In summary, writing polynomials in standard form involves arranging terms by decreasing exponents, combining like terms, and identifying the degree and leading coefficient. Mastery of these concepts is vital for solving polynomial equations and understanding their properties.

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Adding and Subtracting Polynomials

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Adding and Subtracting Polynomials Video Summary

Polynomials are algebraic expressions that consist of variables raised to whole number exponents and their coefficients. When working with polynomials, one of the fundamental operations is adding and subtracting them, which is similar to simplifying algebraic expressions by combining like terms.

To add polynomials, you group and combine like terms. For instance, consider the polynomials $5x^2 + 2x + 3$ and $x^2 + 7x + 8$. You would add the coefficients of the like terms as follows:

For the $x^2$ terms: $5x^2 + x^2 = 6x^2$
For the $x$ terms: $2x + 7x = 9x$
For the constant terms: $3 + 8 = 11$

Thus, the sum of these polynomials is $6x^2 + 9x + 11$, which is expressed in standard form, where terms are arranged in decreasing order of their exponents.

Subtracting polynomials follows a similar process, but it requires careful attention to negative signs. For example, in the expression $3x^2 + 2x + 4 - (5x + 10 - x^2)$, you first distribute the negative sign across the terms in the parentheses:

The $3x^2$ remains unchanged.
The $2x$ remains unchanged.
The $4$ remains unchanged.
Distributing the negative gives $-5x$ and $-10$, and the $-(-x^2)$ becomes $+x^2$.

After distributing, the expression simplifies to $3x^2 + 2x + 4 - 5x - 10 + x^2$. Now, combine the like terms:

For the $x^2$ terms: $3x^2 + x^2 = 4x^2$
For the $x$ terms: $2x - 5x = -3x$
For the constant terms: $4 - 10 = -6$

The final result is $4x^2 - 3x - 6$, also in standard form. It is crucial to be meticulous with the distribution of negative signs during subtraction to avoid errors.

In summary, adding and subtracting polynomials involves combining like terms and ensuring proper handling of negative signs, particularly in subtraction. Mastery of these operations is essential for further studies in algebra and calculus.

Problem

Perform the indicated operation.
$\left(x^3+3x^2-7x\right)+2\left(x^3-5x^2+9x+4\right)$

$2x^3-2x^2+2x+6$

$3x^3-7x^2+11x+8$

$2x^3-2x^2+2x+4$

$3x^3-2x^2+2x+4$

Problem

Perform the indicated operation.
$\left(-2x^4+10x^3+6x-3\right)-\left(x^4-7x^2+8x+5\right)$

$-3x^4+10x^3+7x^2-2x-8$

$-3x^4+17x^3-2x-8$

$-3x^4+17x^2-2x-8$

$-x^4+10x^3-7x^2+14x+2$

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Here’s what students ask on this topic:

What is a polynomial in algebra?

A polynomial in algebra is an algebraic expression consisting of variables and coefficients, where the variables are raised to positive whole number exponents. For example, $x^{3} + 2 x^{2} - 5 x + 7$ is a polynomial. The exponents must be non-negative integers, and the coefficients can be any real numbers. Polynomials are classified based on the number of terms they have: monomials (one term), binomials (two terms), and trinomials (three terms).

How do you write a polynomial in standard form?

To write a polynomial in standard form, arrange the terms in descending order of their exponents. For example, the polynomial $3 x^{2} + 5 x + 4$ is already in standard form because the exponents decrease from 2 to 1 to 0. Ensure all like terms are combined, and the polynomial is simplified. The term with the highest exponent is called the leading term, and its coefficient is the leading coefficient.

How do you add and subtract polynomials?

To add or subtract polynomials, combine like terms, which are terms with the same variable raised to the same power. For example, to add $5 x^{2} + 2 x + 3$ and $x^{2} + 7 x + 8$ , combine the $x^{2}$ terms, the $x$ terms, and the constants: $6 x^{2} + 9 x + 11$ . When subtracting, distribute the negative sign before combining like terms.

What is the degree of a polynomial?

The degree of a polynomial is the highest exponent of the variable in the polynomial. For example, in the polynomial $4 x^{3} + 3 x^{2} - 2 x + 7$ , the highest exponent is 3, so the degree of the polynomial is 3. The degree indicates the polynomial's highest power and helps classify the polynomial's behavior and properties.

What are the leading coefficient and constant term in a polynomial?

The leading coefficient of a polynomial is the coefficient of the term with the highest exponent. For example, in the polynomial $4 x^{3} + 3 x^{2} - 2 x + 7$ , the leading coefficient is 4. The constant term is the term without a variable, which in this case is 7. The leading coefficient and constant term are important for understanding the polynomial's behavior and graph.