Adding Polynomials Horizontally

Summary

This educational video script focuses on teaching the process of polynomial addition using the horizontal method. It demonstrates how to simplify an expression involving two polynomials by keeping the first polynomial as is and then adding the second polynomial's terms without the addition sign and brackets. The script emphasizes grouping like terms together, which are then combined by adding their coefficients while retaining the variable and its exponent. A common mistake is pointed out regarding the handling of signs, particularly with the constant term. The final result is a simplified polynomial in descending order of exponents.

Takeaways

• 📘 The script introduces polynomial addition as one of the first operations to learn in algebra.
• 📐 The horizontal method is a strategy for adding polynomials, which involves solving the problem in a single line.
• 🔍 The first step in the horizontal method is to copy the first polynomial as it is, including its signs.
• ➕ When adding the second polynomial, retain the signs of each term and remove the addition sign and brackets.
• 🔢 Group like terms together, which means combining terms with the same variable and exponent.
• ✏️ Keep the sign of each term with its coefficient when combining like terms.
• 🧮 Combine the coefficients of like terms by adding them together while keeping the variable and its exponent unchanged.
• 🔄 It's important to note that the negative sign in front of a term indicates starting with the negative value and adding to it.
• ⚠️ A common mistake is incorrectly handling negative signs, especially when adding terms with different signs.
• 📉 The final simplified polynomial should be written in descending order of exponents, reflecting the standard form.

Q & A

What is the first strategy mentioned for adding polynomials?

The first strategy mentioned for adding polynomials is the horizontal method, where the entire problem is solved in the same line.

How do you handle the first polynomial when using the horizontal method?

When using the horizontal method, you keep the first polynomial exactly the same as it is shown in the problem, copying it directly.

What is the next step after copying the first polynomial?

The next step is to write the equal sign and then copy the second polynomial, removing the addition sign and brackets but keeping the signs for each term.

Why is it important to keep the signs with the terms when combining polynomials?

It is important to keep the signs with the terms to ensure the correct combination of like terms, maintaining the proper arithmetic operations.

How do you group like terms when simplifying the polynomial?

You group like terms by placing them side by side, remembering to keep their signs, and then combine them by adding their coefficients.

What is a common mistake made when combining the constant terms in a polynomial?

A common mistake is not properly accounting for the signs when combining constants, such as starting with a negative constant and adding a positive one without considering the correct arithmetic.

What is the final step in simplifying a polynomial using the horizontal method?

The final step is to combine the like terms by adding their coefficients and keeping the variable with the exponent the same, resulting in the simplified polynomial.

Why is it necessary to write the simplified polynomial in descending order of exponents?

Writing the simplified polynomial in descending order of exponents is necessary to follow the standard form of polynomials, making it easier to read and understand.

Can you add polynomials using a method other than the horizontal method?

Yes, there are other methods to add polynomials, such as the vertical method, which aligns like terms in columns for easier combination.

What is the significance of the equal sign in the context of adding polynomials?

The equal sign in the context of adding polynomials indicates the starting point for the addition operation and separates the original polynomials that are to be combined.

Outlines

00:00

📚 Polynomial Addition: Horizontal Method

This paragraph introduces the concept of polynomial addition, specifically using the horizontal method. The process begins by identifying two separate polynomials given in a problem statement. The first polynomial is copied directly as shown, and the second polynomial is added to the first by removing the addition sign and brackets while maintaining the signs of the terms. The next step is to group like terms together, which involves aligning terms with the same variable and exponent. Finally, the coefficients of the like terms are combined, and the variable with its exponent remains unchanged. The example provided simplifies the expression (-2d^2 + 6d - 3) and (+5d^2 + 4d + 7) to (3d^2 + 10d + 4), highlighting the importance of correctly handling signs and combining like terms to achieve the simplified polynomial in descending order of exponents.

Keywords

💡Polynomial

A polynomial is a mathematical expression consisting of variables and coefficients, involving only the operations of addition, subtraction, and non-negative integer exponents. In the video, polynomials are the primary objects being manipulated, specifically through the operation of addition. The script describes how to add two polynomials together, which is a fundamental algebraic operation.

💡Addition

Addition is one of the basic arithmetic operations, and in the context of the video, it refers to the process of combining like terms from two polynomials. The script explains how to add polynomials by aligning like terms and then summing their coefficients, which is a standard procedure in algebra.

💡Horizontal Method

The horizontal method is a technique for adding polynomials that involves writing the polynomials side by side and combining like terms horizontally. The video script uses this method as the first strategy to simplify the given expression, showcasing how to align terms with the same variable and exponent and then add their coefficients.

💡Like Terms

Like terms are terms in a polynomial that have the same variable raised to the same power. The script emphasizes the importance of grouping like terms together when adding polynomials, as these are the terms that can be combined directly by adding their coefficients.

💡Coefficients

Coefficients are the numerical factors in a term of a polynomial that multiply the variable. In the video, the script instructs viewers to add the coefficients of like terms while keeping the variable and its exponent unchanged, which is a key step in simplifying polynomials.

💡Variable

A variable is a symbol, often represented by a letter, that stands for an unknown quantity in mathematics. In polynomials, variables are used to represent the unknowns, and the script discusses how to handle variables with their respective exponents when adding polynomials.

💡Exponent

An exponent indicates the number of times a base quantity is multiplied by itself. In the context of polynomials, the exponents show how many times the variable is multiplied by itself. The script mentions that when adding polynomials, the exponents must match for terms to be combined.

💡Simplified Polynomial

A simplified polynomial is the result of combining like terms and reducing a polynomial to its most concise form. The video script demonstrates the process of simplification, leading to a polynomial where no further combination of like terms is possible.

💡Descending Order

Descending order refers to the arrangement of terms in a polynomial from the highest to the lowest power of the variable. The script concludes by noting that the simplified polynomial should be written in descending order of the exponents, which is a standard way to express polynomials.

💡Common Mistake

The script points out a common mistake made when adding constants in polynomials, which is not properly accounting for the signs. It emphasizes the importance of correctly handling signs when combining like terms, especially when constants are involved.

Highlights

Introduction to addition of polynomials, one of the first types of operations learned.

The horizontal method is introduced as a strategy to solve polynomial addition in a single line.

The first polynomial is copied exactly as it appears in the problem.

In an addition problem, all signs for terms in the second polynomial remain the same, and the addition sign and brackets are removed.

The terms are grouped by like terms to prepare for combination.

D^2 terms are grouped together, specifically -2d^2 + 5d^2.

D terms are grouped next, specifically 6D + 4D.

Constants are grouped together, specifically -3 + 7.

Combining terms involves adding the coefficients while keeping the variables with the same exponent unchanged.

Result of combining D^2 terms: -2 + 5 results in 3d^2.

Result of combining D terms: 6D + 4D results in +10D.

Common mistake: misinterpreting the negative sign when combining constants like -3 and 7.

Result of combining constants: -3 + 7 results in a positive 4.

Final simplified polynomial is written in descending order of exponents.

Exponents in the simplified polynomial decrease from the highest (D^2) to the lowest (constant).

Transcripts

one of the first types of operations

you'll learn to perform on polom is

addition and a typical problem will be

worded something like this simplify the

following expression whereby you're

given two different pols and they're

separated by an addition

sign the first strategy we can use to

approach this problem would be called

the horizontal method whereby you'll

solve the entire problem in the same

line so to begin I look at my first

polinomial and and I'm going to just

keep that exactly the same as it is

shown in the problem so put my equal

sign and I'm going to copy out -2d ^2 +

6D - 3 now if I look at my second

polom since this is an addition problem

I can keep all of the signs for the

different terms the same all I have to

do is lose the addition sign and the

brackets and then keep the signs for

each different term what that gives me

is plus 5 d^

2 + 4 d + 7 okay the next step is to

group The like terms together so we can

begin to combine them so let's take our

D2 terms and put them side by side so

remember keep your sign with your term

so -2d 2 + 5

d^2 that takes these two terms they're

covered now we have +

6D

+

4D -

3 plus

7 the final step is going to involve

combining these terms together so -2 + 5

is going to be 3 d^2 we're just adding

the co coefficients and keeping our

variable with the exponent the same 6D +

4D equal POS 10 D and here's a common

mistake that's often made the negative

sign right here means we are starting

with theg -3 and we're adding seven to

that so -3 + 7 is going to give us a

positive 4 and this is our simplified

polinomial written in descending order

you notice that the exponent are

decreasing as we

go