Multiplying Rational Expressions
Summary
TLDRThis educational video script offers a comprehensive guide on multiplying and simplifying rational expressions. It begins with a step-by-step demonstration of factoring and canceling terms in a given expression, highlighting the importance of identifying and extracting the greatest common factor (GCF). The script then moves on to factoring trinomials and using the difference of squares technique. It further illustrates how to handle expressions with leading coefficients other than one by multiplying them with the constant term. The video also addresses the identification of excluded values, crucial for avoiding division by zero. Finally, it demonstrates the process of factoring a difference of perfect cubes and concludes with a complete factorization and simplification of a complex rational expression.
Takeaways
- π Multiplying rational expressions involves converting them into fractions and then multiplying the numerators and denominators accordingly.
- π Factoring expressions completely is crucial before multiplying to simplify the process and identify common factors for cancellation.
- 𧩠The greatest common factor (GCF) is used to simplify expressions by dividing out common factors from the numerator and denominator.
- βοΈ Cancelling common factors between the numerator and denominator is a key step in simplifying rational expressions.
- π’ Factoring trinomials involves finding two numbers that multiply to the constant term and add up to the middle coefficient.
- π The difference of squares technique is used to factor expressions of the form a^2 - b^2 into (a + b)(a - b).
- π Factoring by grouping is a method used for expressions where the coefficients follow a pattern that allows for common factors to be extracted.
- π Identifying excluded values, or points of discontinuity, is important as they represent values that make the denominator zero and are not valid for the expression.
- π The difference of cubes formula a^3 - b^3 = (a - b)(a^2 + ab + b^2) is used to factor expressions involving the difference of two cubes.
- β Simplifying rational expressions results in a final expression that is easier to work with and understand.
Q & A
What is the first step when multiplying rational expressions?
-The first step is to ensure that the expressions are in fractional form and then to factor them completely.
How do you simplify the expression 7x + 14 divided by x^2 - 8?
-First, factor out the greatest common factor (GCF) of 7 from the numerator to get 7(x + 2). Then factor the denominator using the difference of squares to get (x + 2)(x - 2). After canceling out common factors, the simplified form is x + 2.
What technique is used to factor x^2 - 4?
-The difference of perfect squares technique is used, which results in (x + 2)(x - 2).
What are the excluded values when simplifying rational expressions?
-Excluded values are the values of the variable that make any denominator zero. These values are identified by setting each factor in the denominator equal to zero.
How do you factor the trinomial 10x^3 - 70x^2 + 120x?
-First, take out the GCF of 10x, resulting in x^2 - 7x + 12. Then, factor by grouping or by finding two numbers that multiply to 24 (2*12) and add to -7 (-8 + 3), which are -8 and 3. The factored form is (x - 3)(x + 4).
What is the process for factoring a difference of perfect cubes?
-The formula for factoring a difference of perfect cubes is a^3 - b^3 = (a - b)(a^2 + ab + b^2). Identify 'a' and 'b' such that a^3 is the first term and b^3 is the last term of the cube difference.
How do you simplify the expression 5x^2 - 15x divided by 2x^2 + 11x + 12?
-First, take out the GCF of 5x from the numerator. Then, factor the quadratic in the denominator by finding two numbers that multiply to 24 and add to 11, which are 8 and 3. The simplified form is x - 3 over 2(x + 4).
What is the final simplified form of the expression 7x + 14 divided by x^2 - 8 multiplied by x^2 + 3x - 10?
-After factoring and canceling, the final simplified form is 7(x + 5)/2.
How do you identify the points of discontinuity in a rational expression?
-Points of discontinuity are identified by finding the values of the variable that make any factor in the denominator equal to zero, as these values would make the expression undefined.
What is the final simplified form of the expression 3x^3 - 24 divided by 2x^2 - 14x + 20 times 4x^3 - 20x^2 + 3x - 15 divided by x^2 + 6x + 12?
-After factoring and canceling, the final simplified form is 4x^2 + 3/2.
Outlines
π Multiplying and Simplifying Rational Expressions
This paragraph introduces the process of multiplying and simplifying rational expressions. The first example involves multiplying \( \frac{7x+14}{x^2-8} \) by \( \frac{x^2+3x-10}{1} \). The process starts by factoring the expressions and identifying the greatest common factor (GCF). The GCF of 7x+14 is 7, which simplifies to \( \frac{x+2}{x^2-4} \). The denominator \( x^2-4 \) is factored using the difference of squares technique into \( (x+2)(x-2) \). The numerator \( x^2+3x-10 \) is factored into \( (x+5)(x-2) \). After canceling common factors, the final simplified form is \( \frac{7(x+5)}{2} \). The paragraph also explains how to identify excluded values, which are the values that make the denominator zero and thus are not in the domain of the function.
π Factoring Techniques and Simplifying Expressions
The second paragraph delves into more examples of multiplying and simplifying rational expressions. It begins with the expression \( \frac{5x^2-15x}{2x^2+11x+12} \times \frac{3x^2-48}{10x^3-70x^2+120x} \). The process involves factoring out the GCFs and then simplifying the expressions. The numerator \( 5x^2-15x \) simplifies to \( x(3x-3) \) after factoring out the GCF of 5x. The denominator \( 2x^2+11x+12 \) is factored by grouping, resulting in \( 2(x+4)(x+3) \). The second part of the expression is simplified by factoring out the GCF of 10x and then factoring the resulting trinomial. The final simplified form after canceling common factors is \( \frac{3}{2(2x+3)} \). The paragraph also discusses how to identify excluded values by setting factors equal to zero and solving for x.
π Advanced Factoring and Canceling in Rational Expressions
The third paragraph presents a more complex example of multiplying and simplifying rational expressions, involving the expression \( \frac{3x^3-24}{2x^2-14x+20} \times \frac{4x^3-20x^2+3x-15}{x^2+6x+12} \). The process starts by factoring out the GCFs and then using advanced factoring techniques such as the difference of cubes and factoring by grouping. The numerator \( 3x^3-24 \) simplifies to \( x(x^2-8) \) after factoring out the GCF of 3. The denominator \( 2x^2-14x+20 \) simplifies to \( 2(x^2-7x+10) \) by dividing each term by 2. The expression \( x^2-7x+10 \) is then factored by grouping into \( (x-5)(x-2) \). After canceling common factors, the final simplified form is \( \frac{4x^2+3}{2} \). The paragraph also emphasizes the importance of canceling all possible factors and identifying excluded values to ensure the solution is fully simplified and valid.
Mindmap
Keywords
π‘Rational Expressions
π‘Factoring
π‘Greatest Common Factor (GCF)
π‘Simplifying
π‘Difference of Perfect Squares
π‘Trinomial
π‘Canceling
π‘Factor by Grouping
π‘Excluded Values
π‘Difference of Perfect Cubes
Highlights
Introduction to multiplying and simplifying rational expressions
Example of multiplying 7x + 14 by x^2 + 3x - 10
Factoring out the greatest common factor (GCF) of 7 from 7x + 14
Simplifying 2x^2 - 8 by factoring out 2
Factoring the trinomial x^2 - 4 into (x + 2)(x - 2)
Cancellation of common factors (x + 2) and (x - 2) in the expression
Final simplified form of the first example as 7(x + 5) / 2
Second example involving 5x^2 - 15x and 3x^2 - 48
Factoring out the GCF of 5x from 5x^2 - 15x
Factoring 3x^2 - 48 using the difference of squares technique
Factoring the cubic polynomial 10x^3 - 70x^2 + 120x
Identifying the excluded values for the rational expression
Third example with 3x^3 - 24 and 2x^2 - 14x + 20
Factoring by grouping in the expression 4x^3 - 20x^2 + 3x - 15
Using the formula for the difference of perfect cubes to factor x^3 - 8
Final simplified form of the third example as 4x^2 + 3 / 2
Transcripts
in this video we're going to focus on
multiplying rational expressions
and simplifying them as well
so here's the first example
7x plus 14
divided by
x squared
minus eight
and let's multiply it by
x squared
plus three x
minus ten
feel free to try this example
now the first thing i would do is
put the second part into a fraction
write it over one
now what you want to do is you want to
factor completely and cancel
7x plus 14 has a gcf of 7 so we want to
take that out
7x divided by 7 is x
14 divided by 7 is 2.
now 2x squared minus 8 we can also take
out the greatest common factor which is
2
and that's going to leave behind x
squared and negative eight divided by
two is negative four
on the right we have a trinomial with a
leading coefficient of one
so we need to find two numbers that
multiply to negative ten but add to
three
so that's positive five and negative two
so this is going to be x plus five
and x minus two
now let's factor x squared minus four
using the difference of perfect squares
technique we know it's going to be
the square root of x squared is x the
square root of 4 is 2
and so it's going to be x plus 2 and x
minus 2.
now let's see what we can cancel
so we can cancel an x plus two
and we can cancel an x minus two
so we have a seven left over
and x plus five
and a two
so the final answer is let's see if i
can fit it in here
seven
times x plus five
divided by two
so that's the solution to this problem
fully simplified
now let's try another example
5x squared
minus 15x
divided by
2x squared
plus 11x plus 12
multiplied by
3x squared minus 48
divided by
10 x cubed
minus 70
x squared
plus
120x
so go ahead and simplify factor
everything
and cancel
so 5x squared minus 15x we can take out
the gcf
which is 5x
5x squared divided by 5x is x
negative 15x divided by 5x
is negative 3.
now 3x squared minus 48
we can take out a 3
which will give us x squared minus 16.
and
we can factor that using the difference
of squares technique
the square root of 16 is 4.
so we're going to have x plus 4 and x
minus four
now what about ten x cubed
minus seventy x squared plus one twenty
x
well we can begin by taking out the gcf
which is 10x
so we're going to have x squared
negative 70x squared divided by 10x
is negative 7x
120x divided by 10x is 12.
two numbers that multiply to 12 but add
to negative seven
are negative three and negative four
so this is going to be x minus three
and x minus four so let's go ahead and
put that here
and so this is what we have
let's just get rid of this
now let's focus on the last part
2x squared plus 11x
plus 12.
we have a trinomial with the leading
coefficient being something other than
one it's two
so when it's not one
multiply the leading coefficient by the
constant term
so two times 12 which is 24.
now what two numbers multiply to 24
but add to the middle coefficient 11.
this is going to be 8 and 3. 8 plus 3 is
11. 8 times 3 is 24. so we're going to
do is
we're going to replace 11x
with 8x plus 3x
and then factor by grouping
so in the first two terms take out the
gcf
the gcf is 2x
and so we'll be left with x plus four
and in the last two terms take out the
greatest common factor
which is three
and we're going to get x plus four
so we're gonna have two parentheses one
of which will contain x plus four
and the other one it's going to have the
stuff on the outside the two x plus
three
and so that's how we can factor it
so it's x plus four times two x plus
three
so now let's simplify
what can we cancel
we can cancel an x minus three
and we can cancel an x minus four
in addition to that we can get rid of x
plus four
and also we can take out an x
and also we can reduce five over ten
five over ten reduces to one over two
ten divided by five is two
so what we have left over
is a three
a two and a two x plus three
so the final answer is three divided by
two times
two x plus three
and so that is the solution
sometimes you need to identify
the excluded values
and here's the basic idea
you cannot have a zero in the
denominator
so four
will produce a zero in the denominator
even though it cancels and so will three
zero if you just see x
negative four
and also if you set two x plus three
equal to zero
you should get negative three over two
so that's how you can identify the
excluded values so we have negative 4
and 4
0 and 3 and negative 3 over 2.
those are the points of discontinuity
x cannot equal any of those numbers
now let's work on another problem
3x q minus 24
divided by
2x squared
minus 14x
plus
20. times 4x cubed
minus 20x squared
plus
3x
minus 15
divided by
x squared
plus six x plus twelve
so pause the video try this problem
now the first thing we need to do is
look for any gcfs
and notice that both numbers on top are
divisible by three so we can take out a
three
three x cubed divided by three is x
cubed
negative 24 divided by three is negative
eight
now in the denominator we all have even
numbers which means that we can divide
each of those by two
and so we'll be left with x squared
minus seven x plus ten
here all the coefficients are divisible
by three
so if we take out a three it's going to
be x squared plus
two x
plus four
now for this expression
notice that we can factor it by grouping
negative 20 divided by 4 is negative 5.
negative 15 divided by 3 is also
negative 5.
so if the first two coefficients have
the same ratio as the last two
coefficients
you can factor by grouping
so let's take out the gcf in the first
two terms
which is going to be 4x squared
4x cubed divided by 4x squared is x
negative 20x divided by 4x squared
is negative 5.
now in the last two terms
we can take out a 3.
3x divided by 3 is x
negative 15 divided by 3 is
negative 5.
now let's start with this
how can we factor x cubed minus eight
here's the formula that we need to use
a to the third minus b to the third
is going to be a minus b
times a squared
plus a b
plus a b squared
so if a to the third
is x to the third
a has to be x
and if b to the third is eight b has to
be the cube root of eight which is two
so a is x b is two so a squared is going
to be x squared
and then a b that's
x times negative two
well actually b is positive 2.
this negative sign is already there so
a times b is going to be x times 2 which
is positive 2x
there's a plus sign
and then b squared that's 2 times 2
which is 4.
so that's how you can factor
a difference of
perfect cubes
now let's factor the trinomial two
numbers that multiply to 10 but add to
negative seven
that's going to be negative two and
negative five
so this becomes x minus five
and x minus two
now we can't really factor
x squared plus two x plus four but we
don't need to because it will be
canceled soon
so i'm just gonna rewrite it for now
now going back to this expression
we need to take out the x minus five
since it's common to both terms
and
we're gonna have four x squared plus 3
left over
so now the rational expression is
completely factored
now let's cancel
so we can cancel x minus 2.
we can cancel x squared plus two x plus
four
we can cancel x minus five
and
we can cancel a three
so the only things that we have left
over
are those two so the final answer is 4x
squared plus 3
divided by 2
and that's it
you
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