Polynomials - Adding, Subtracting, Multiplying and Dividing Algebraic Expressions
Summary
TLDRThis 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
📘 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.
🔢 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.
📚 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.
🔄 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.
Mindmap
Keywords
💡Polynomial Expressions
💡Like Terms
💡Distributive Property
💡FOIL Method
💡Binomial
💡Trinomial
💡Factoring
💡Long Division (Polynomials)
💡Synthetic Division
💡Combining 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
check out my channel
um you can find my playlist on my
website or on my channel
so if you like this video feel free to
subscribe
and uh thanks for watching
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