Polynomials - Adding, Subtracting, Multiplying and Dividing Algebraic Expressions

Summary

This educational video script offers a comprehensive guide on polynomial operations, including addition, subtraction, and multiplication. It demonstrates how to combine like terms and use methods like FOIL for binomial multiplication. The script also covers more complex scenarios like multiplying binomials by trinomials and dividing polynomials using factoring, long division, and synthetic division. The presenter encourages viewers to practice these techniques and directs them to additional resources for further learning in various subjects.

Takeaways

• 🔢 To add polynomial expressions, combine like terms by adding their coefficients.
• ➖ When subtracting polynomials, distribute the negative sign to each term in the second polynomial and then combine like terms.
• 🔄 When multiplying binomials, use the FOIL method (First, Outer, Inner, Last) to multiply and then combine like terms.
• 📚 For multiplying a binomial by a trinomial, expect to initially have six terms before combining like terms.
• 🔗 When multiplying polynomials, always double-check your work to ensure accuracy.
• 📉 To divide polynomials, consider factoring, long division, or synthetic division methods.
• ✂️ Factoring involves finding two numbers that multiply to the constant term and add to the linear coefficient.
• 🔄 Long division of polynomials is similar to long division of numbers, with the division symbol placed outside the dividend.
• 🔄 Synthetic division is a shortcut for dividing polynomials when the divisor is a linear term, using the root of the divisor.
• 📈 The video provides a comprehensive guide to polynomial operations, including addition, subtraction, multiplication, and division.

Q & A

What is the first step when adding polynomial expressions?

The first step when adding polynomial expressions is to combine like terms. Like terms are terms that have the same variable raised to the same power.

How do you combine like terms in the example given in the video?

In the example, 4x^2 and 3x^2 are like terms, which combine to 7x^2. 5x and -8x combine to -3x. The constants 7 and 12 combine to 19.

What is the process for subtracting polynomial expressions as described in the video?

To subtract polynomial expressions, distribute the negative sign to every term in the second polynomial, change the signs of those terms, and then combine like terms.

How does the video demonstrate the multiplication of two binomials?

The video demonstrates the multiplication of two binomials using the FOIL method (First, Outer, Inner, Last), which involves multiplying each term in the first binomial by each term in the second binomial and then combining like terms.

What is the result of multiplying (3x + 5) by (2x - 3) as shown in the video?

The result of multiplying (3x + 5) by (2x - 3) is 6x^2 + x - 15 after applying the FOIL method and combining like terms.

How does the video simplify the expression (2x - 5)^2?

The video simplifies (2x - 5)^2 by applying the FOIL method to two binomials (2x - 5) multiplied by each other, resulting in 4x^2 - 20x + 25.

What is the initial step when multiplying a binomial by a trinomial according to the video?

The initial step when multiplying a binomial by a trinomial is to distribute each term in the binomial to each term in the trinomial, resulting in six terms before combining like terms.

How does the video approach the division of polynomials?

The video approaches the division of polynomials by suggesting three methods: factoring, long division, and synthetic division.

What is the result of dividing x^2 + 7x + 12 by x + 3 using factoring as shown in the video?

Using factoring, x^2 + 7x + 12 can be factored into (x + 3)(x + 4), so when divided by x + 3, the result is x + 4.

How does the video use synthetic division to divide 2x^2 - 7x + 6 by x - 2?

The video uses synthetic division by writing the coefficients of the numerator (2, -7, 6) and using the root x = 2 to perform the division, resulting in a quotient of 2x - 3 and a remainder of 0.

Outlines

00:00

📘 Polynomial Operations Overview

This paragraph introduces the basics of adding, subtracting, and multiplying polynomial expressions. It explains the process of combining like terms when adding polynomials, such as adding 4x squared and 3x squared to get 7x squared. It also demonstrates how to subtract polynomials by distributing the negative sign and combining like terms, as shown in the example of subtracting (5x squared - 7x + 13) from (9x squared - 7x + 13). The process involves changing the signs of the terms on the right and then combining like terms to simplify the expression.

05:00

🔢 Advanced Polynomial Operations

This section delves into more complex polynomial operations, including distributing coefficients across terms and combining like terms. It illustrates the process with an example where a polynomial with coefficients is multiplied out and simplified. The paragraph also covers the FOIL method for multiplying binomials and extends the concept to multiplying a binomial by a trinomial, resulting in six terms before combining like terms. The importance of double-checking work is emphasized to ensure accuracy in polynomial multiplication.

10:02

📚 Multiplication and Division of Polynomials

The paragraph focuses on multiplying and dividing polynomials. It starts with multiplying a trinomial by another trinomial, resulting in nine terms before combining like terms. The process involves multiplying each term of one polynomial by each term of the other and then combining like terms to simplify the result. The paragraph also discusses the division of polynomials, introducing methods like factoring, long division, and synthetic division. Examples are provided for each method, demonstrating how to simplify expressions by canceling out common factors or using division techniques.

15:02

🔄 Polynomial Division Techniques

This final paragraph emphasizes the division of polynomials, particularly using long division and synthetic division. It provides a step-by-step guide on how to perform long division with polynomials, including setting up the division, performing the division, and finding the remainder. Synthetic division is also explained, showing how to use it to simplify the division process. The paragraph concludes with a reminder to check out the video creator's website and channel for more educational content on various subjects.

Keywords

💡Polynomial Expressions

Polynomial expressions are mathematical expressions consisting of variables and coefficients, involving operations of addition, subtraction, and multiplication. In the video, the instructor demonstrates how to add, subtract, and multiply polynomials by combining like terms and using distributive properties. For example, expressions like '4x² + 5x + 7' and '3x² - 8x + 12' are combined by adding like terms.

💡Like Terms

Like terms in algebra are terms that have the same variable raised to the same power. These terms can be combined through addition or subtraction. In the video, like terms such as '4x²' and '3x²' are combined by adding their coefficients, resulting in '7x²'. Identifying and combining like terms is crucial when simplifying polynomial expressions.

💡Distributive Property

The distributive property allows for the multiplication of a single term across terms within parentheses. In the video, this property is used when distributing a negative sign or a constant across terms in a polynomial. For example, in '4(3x² + 6x - 8)', the 4 is distributed to each term to produce '12x² + 24x - 32'.

💡FOIL Method

The FOIL (First, Outer, Inner, Last) method is a technique for multiplying two binomials. It involves multiplying the first terms, outer terms, inner terms, and last terms, and then combining like terms. In the video, the instructor uses FOIL to multiply '(3x + 5)' and '(2x - 3)', producing '6x² + 1x - 15'.

💡Binomial

A binomial is a polynomial with exactly two terms, typically connected by addition or subtraction. The video uses binomials such as '3x + 5' and '2x - 3' when teaching multiplication of polynomials. The instructor also explains how binomials are involved in operations like FOIL and factoring.

💡Trinomial

A trinomial is a polynomial expression that consists of three terms. The video demonstrates how to multiply a binomial by a trinomial, such as '4x - 2' multiplied by 'x² + 3x - 5'. After multiplying the terms, the result has six terms before like terms are combined.

💡Factoring

Factoring is the process of expressing a polynomial as the product of simpler polynomials. In the video, the instructor factors the trinomial 'x² + 7x + 12' as '(x + 3)(x + 4)' to divide it by another polynomial. Factoring is presented as an alternative to long division or synthetic division.

💡Long Division (Polynomials)

Long division is a method used to divide polynomials, similar to the long division process for numbers. In the video, the instructor uses long division to divide polynomials like '2x² - 7x + 6' by 'x - 2', systematically reducing the polynomial step by step until the remainder is zero.

💡Synthetic Division

Synthetic division is a shortcut method for dividing a polynomial by a binomial of the form 'x - c'. The instructor demonstrates synthetic division by dividing '2x² - 7x + 6' by 'x - 2' and simplifies the division process using coefficients and substitution. This method is quicker than long division but only applies to certain types of division problems.

💡Combining Like Terms

Combining like terms involves adding or subtracting terms in a polynomial that have the same variable raised to the same power. In the video, this concept is repeatedly used when simplifying expressions. For instance, in '9x² - 5x²', the result is '4x²' after combining the like terms.

Highlights

Introduction to adding, subtracting, and multiplying polynomial expressions.

Combining like terms to add polynomials, demonstrated with 4x^2 + 5x + 7 and 3x^2 - 8x + 12.

Subtraction of polynomials by distributing the negative sign and combining like terms.

Example of polynomial subtraction: (9x^2 - 7x + 13) - (5x^2 - 7x - 14).

Combining like terms results in 4x^2 + 27 after subtraction.

Distributing numbers in front of polynomials during multiplication.

Multiplying polynomials using the FOIL method for binomials.

Simplifying expressions with squared binomials, such as (2x - 5)^2.

Multiplying a binomial by a trinomial results in six terms before combining like terms.

Example of multiplying (4x - 2)(3x^2 + 6x - 5).

Combining like terms after multiplying binomials and trinomials.

Distributing and combining terms when multiplying trinomials, such as (3x^2 - 5x + 7)(2x^2 + 6x - 4).

Dividing polynomials by factoring, long division, or synthetic division.

Example of dividing (x^2 + 7x + 12) by (x + 3) using factoring.

Long division method demonstrated with (2x^2 - x + 6) divided by (x - 2).

Synthetic division method applied to divide (2x^2 - 7x + 6) by (x - 2).

Final results of polynomial division using different methods.

Encouragement to explore more videos on algebra, trigonometry, precalculus, chemistry, and physics.

Transcripts

in this video we're going to talk about

how to add

subtract

and multiply polynomial expressions

so let's begin

let's say if we have 4x squared

plus five x

plus seven

plus

three x squared minus eight x

plus twelve

so how can we add these two polynomial

expressions

if you know what to do feel free to

pause the video and work out this

particular example

what we need to do is combine like terms

4x squared and 3x squared are like terms

so let's add them 4 plus 3 is 7

so this is going to be 7x squared

now 5x and negative 8x are like terms

5 minus 8

is negative 3

and finally we can add 7 and 12

which together is 19.

so that wasn't too bad right let's try

another example

go ahead and try this one

nine x squared

minus seven x

plus thirteen

minus five x squared

minus seven

x and minus 14.

so go ahead and subtract these two

polynomial expressions

now the first thing i would do is

distribute the negative sign to every

term on the right the signs will change

on the left side you can just open the

parenthesis if there's no number in

front of it you can just rewrite it as

9x squared

minus 7x

plus 13.

and then if we distribute the negative

sign to the other three terms

it's going to be negative five x squared

plus

seven x

plus fourteen

and now let's combine like terms

so we can combine those two

nine minus five is four

so it's four x squared

negative seven x plus seven x is zero so

they will cancel and thirteen plus

fourteen

is uh 727

so this is the answer 4x squared plus

27.

so here's another

problem that we can work on

3x cubed minus five x plus eight

minus

seven x squared

plus six x

minus nine

so let's distribute the negative sign

just like we did before

so the first three terms will remain the

same and then we'll have negative seven

x squared

minus six x

plus nine

so now let's go ahead and combine like

terms

so there's no similar term

to three x cubed there's only one x

cubed term

so we're just gonna

bring it down and rewrite it

likewise this term is one of a kind

so we're just going to rewrite it

now we can combine these two terms

negative 5 minus 6

is negative 11

and 8 plus 9

is 17.

so this is the answer 3x cubed

minus 7x squared minus 11x

plus 17.

now what if we had numbers in front

what would you do in this case

so the first thing we should do

is distribute the four

to these three terms

so four

times 3x squared

is 12x squared

and then 4 times 6x

that's equal to 24x

and 4 times negative 8

is negative 32.

now let's distribute the negative 3 to

the 3 terms on the right negative 3

times 2x squared

is negative

x squared negative three times negative

five x

is positive fifteen x

and finally negative three times seven

is negative twenty one

so now let's combine like terms

twelve minus 6

is positive 6

24 plus 15

is 39

negative 32 minus 21

is negative 53.

so this is it

now let's talk about how to multiply

polynomial expressions

let's start with two binomials

so let's say if we have three x plus

five

multiplied by

two x minus three

we need to use the foil method

three x times two x

is six x squared

three x times negative three is negative

nine x

five times two x

is ten x

and finally five times negative three

is negative fifteen

so now at this point we can combine like

terms negative nine plus ten

is positive one

the other two terms we can bring it down

so it's going to be six x squared plus

one x

minus fifteen

so that's what you can do in order to

multiply two binomials together

now what if you were to see an

expression that looks like this

two x minus five squared

how can you simplify this expression

if you see something like this

this simply means that you have two

binomials multiplied to each other

so there's two 2x minus fives

so let's do what we did in the last

example let's foil

2x times 2x

is equal to 4x squared

2x times negative 5 is negative 10x

negative 5 times 2x

is also negative 10x and finally

negative 5 times negative 5

is positive 25. so now let's combine

these terms

negative 10x minus 10x is negative 20x

and so this is the answer

it's 4x squared minus 20x plus 25.

now what if we want to multiply

let's say a binomial

by a trinomial

how can we do so now notice that when we

multiply a binomial with another

binomial

that is an expression with two terms by

another expression with two terms

initially we got four terms before we

added

like terms

now in this example we have a binomial

which contains two terms and a trinomial

which has three two times three is six

so when we multiply before we combine

like terms we should have uh six terms

so let's go ahead and multiply

4x times x squared

is 4x cubed

4x times 3x

is

x squared

4x times negative 5

is negative 20x

negative 2 times x squared

is negative 2x squared

negative 2 times 3x

is negative 6x

and negative 2 times negative 5

is positive 10.

so let me just double check and make

sure that i didn't make any mistakes

so i believe everything is good now

let's go ahead and combine like terms

it's always good to double check your

work

so this term is one of a kind

so let's simply rewrite it

these two

are like terms 12 minus 2 is 10

and these two are like terms

negative 20 minus 6 is

negative 26 x

plus 10. but as you can see before we

combine like terms

notice that we have a total of six terms

initially

anytime you multiply a binomial by a

trinomial

you will initially get six terms

what's going to happen if we multiply

a trinomial

by another trinomial

go ahead and try it

so 3 times 3 is 9. initially before we

combine like terms we should have 9

terms

so 3x squared times 2x squared

is 6 x to the fourth power

and then 3x squared

times 6x

that's going to be 18

3 times 6 is 18. x squared times x is x

cubed

and then 3x squared times negative 4 is

simply negative 12x squared

next we have negative 5x times 2x

squared

that's

negative 10x cubed

and then negative 5x times six x

which is negative thirty x

and negative five x times negative four

wait negative five x times six x is

negative thirty x squared

it's always good to double check the

work negative 5x times negative 4 is 20x

and then

7 times 2x squared

that's going to be 14

x squared

and then 7 times 6x

is positive 42x

and finally 7 times negative 4

is negative 28.

so i'm just going to take a minute and

double check everything make sure

i didn't miss anything

so i believe everything is correct up to

this point

so as you can see

we have nine terms at this point

now let's go ahead and combine like

terms

so we have six x to the fourth

and we can combine these two

eighteen minus ten is positive eight

and

there's three terms with an x squared

attached to it

negative twelve plus 14

is positive 2

and positive 2 minus 30

is negative 28

now we have these two terms to add 42

plus 20 is 62

and then the last term

so this is it 6x to the fourth plus 8x

cubed

minus 28x squared plus 62x

minus 28. so now you know how to

multiply a trinomial with another

trinomial

now what about dividing polynomials

let's say if

we wish to divide the trinomial x

squared plus seven x

plus fifteen

actually instead of plus fifteen let's

say plus twelve

let's divide it by x plus three how can

we do so

there's three things that you can do

you can factor you can use long division

or you can use synthetic division

let's divide by factoring

to factor the trinomial

we need to find two numbers that

multiply to twelve

but add to seven

three times four

is twelve three plus four is seven so we

can factor it like this it's x plus

three times x plus four

now we can cancel these two uh terms

so therefore it's x plus four so x

squared plus seven x plus 12 divided by

x plus three is x plus four

so that's how you can divide two

polynomial expressions

um by factoring

just factor and cancel

now let's try another example

2x squared minus x

plus 6

divided by x minus 2.

now you can factor the numerator it is

factorable and you can cancel so you can

use the other method as well but for

this particular example let's use long

division

so i'm going to put the denominator on

the outside

and the numerator

on the inside

so first

we're going to divide 2x squared by x

2x squared divided by

x is 2x

now we're going to multiply 2x times x

is 2x squared

and two x times negative two

is negative four x

and now subtract two x squared minus two

x squared is zero

so those two cancel

and then negative one x

minus negative four x is the same as

negative one x plus four x

which is positive three x

six minus nothing or six minus zero

is simply six

so we can bring the six down

now let's try another example

let's divide two x squared

minus seven x plus six

by

x minus 2.

now the numerator is factorable

but

we're going to use synthetic division

and long division

you can factor and cancel if you want

but let's start with long division

let's put the denominator on the outside

and the numerator on the inside

so first let's divide

2x squared divided by x

is simply 2x

so now let's multiply

two x times x

is two x squared

two x times negative two

is negative four x

and now we're going to subtract

2x squared minus 2x squared

is 0 they cancel

negative 7x minus negative 4x

which is the same as negative 7x plus 4x

that's negative three x

and six minus nothing or six minus zero

is simply six

so we can bring the six down

so now let's divide

negative three x divided by x

is negative 3.

and now let's multiply negative 3 times

x

is negative 3x

and negative 3 times negative 2

is positive 6.

so now let's subtract negative three x

minus negative three x or negative three

x plus three x is zero six minus six is

zero

so the remainder is zero therefore

this is equal to two x minus three

so that's how you can divide

polynomial expressions using long

division

now let's see if we can get the same

answer using synthetic division

let's write the coefficients

of the numerator which are two negative

seven and six

now we're dividing it by x minus two

if you set this equal to zero

x is two

so we're going to use two here instead

of negative two

let's bring down the two

two times two

is four

and negative seven plus four is negative

three

so you gotta multiply add multiply add

and so forth two times negative three

is negative six

and six plus negative six is zero

so this is the remainder

negative three

is the constant

and two has the x with it so it's two x

minus three

when you divide 2x squared by x

you're going to get 2x

so the first term is x to the first

power

so you can divide polynomials by

factoring by using long division or

synthetic division so that is it for

this video thanks for watching

if you want to find more videos on

algebra trig precal chemistry physics

check out my website video.tutor.net or

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