ILLUSTRATING POLYNOMIAL FUNCTIONS || GRADE 10 MATHEMATICS Q2
Summary
TLDRThis video lesson explains the fundamentals of polynomial functions, focusing on how to identify the degree, leading coefficient, and constant term. The instructor defines polynomial functions, provides examples in standard form, and walks through arranging terms in descending order of exponents. The lesson also covers determining whether a given function is polynomial or not, expanding factored polynomials using methods like FOIL, and identifying leading terms and coefficients. Viewers are guided step-by-step through various examples to solidify their understanding of these key concepts.
Takeaways
- đ A polynomial function is defined as a function of the form p(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 where a_n â 0.
- đ The term a_nx^n is known as the leading term of the polynomial.
- đ The coefficient a_n is referred to as the leading coefficient.
- đ The degree of a polynomial is determined by the highest power of x present in the polynomial.
- đą Constant term a_0 is the term without the variable x.
- đ Polynomial functions can be represented in various notations such as p(x), f(x), or y.
- đ Polynomials are written in standard form with terms arranged in decreasing order of exponents.
- đ« Polynomials do not include negative exponents, fractions, or radicals in the variable's denominator.
- đ To identify if a function is a polynomial, check for the absence of variables in denominators, negative exponents, and radicals.
- đ Examples are provided to demonstrate how to write polynomials in standard form and identify their leading term, leading coefficient, and degree.
- đ The script concludes with a prompt to like, subscribe, and hit the bell button for more educational content.
Q & A
What is a polynomial function?
-A polynomial function is a function of the form p(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0, where a_n â 0 and n is a non-negative integer. The coefficients a_0, a_1, ..., a_n are real numbers.
What is the leading term of a polynomial function?
-The leading term of a polynomial function is the term with the highest exponent of the variable, which is a_n x^n in the polynomial p(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0.
What is the leading coefficient of a polynomial function?
-The leading coefficient is the coefficient of the leading term, which is a_n in the polynomial p(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0.
What is the constant term of a polynomial function?
-The constant term of a polynomial function is the term without the variable, which is a_0 in the polynomial p(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0.
How do you identify the degree of a polynomial function?
-The degree of a polynomial function is the highest exponent of the variable in the polynomial. It is determined by the leading term.
What is the standard form of a polynomial function?
-The standard form of a polynomial function is when the terms are arranged in decreasing order of the exponents of the variable, starting from the highest exponent to the constant term.
Can a polynomial function have a negative exponent?
-No, a polynomial function cannot have a negative exponent. The exponents in a polynomial function are non-negative integers.
Can a polynomial function have a variable in the denominator?
-No, a polynomial function cannot have a variable in the denominator. The variable must only appear in the numerator and raised to non-negative integer powers.
What is the difference between a polynomial function and a rational function?
-A polynomial function consists of terms with non-negative integer exponents and no variables in the denominator, while a rational function is a ratio of two polynomial functions and can have variables in the denominator.
How do you determine if a given function is a polynomial?
-A given function is a polynomial if it meets the criteria: it has non-negative integer exponents, no variables in the denominator, no fractional exponents, and no radical signs involving variables.
What are the steps to write a polynomial function in standard form?
-To write a polynomial function in standard form, you need to arrange the terms in decreasing order of the exponents of the variable, combine like terms, and ensure that the function does not contain any negative exponents, fractional exponents, radical signs, or variables in the denominator.
Outlines
đ Introduction to Polynomial Functions
In this paragraph, the instructor introduces the concept of polynomial functions, explaining its basic components such as degree, leading coefficient, and constant term. The formula of a general polynomial function is provided in the form P(x) = a_n x^n + a_{n-1}x^{n-1} + ... + a_1 x + a_0, with an emphasis on the leading term, leading coefficient, and constant term. The speaker highlights how different notations such as P(x), F(x), and Y can represent polynomial functions.
âïž Writing Polynomial Functions in Standard Form
The second paragraph focuses on arranging polynomial functions in standard form, where terms are ordered by decreasing powers of the variable. Several examples, including F(x) = 4x^3 - 16x - 4 + x^4 - x^2, are rewritten in standard form, emphasizing the importance of arranging terms based on their exponents. Additional examples, such as Y = 1/6 x^4 - x^2 + 5x^5 + 7x^3 - 5, further illustrate the rearrangement process.
đ Identifying Polynomial and Non-Polynomial Expressions
This paragraph explains how to determine whether a function is a polynomial or not. The key restrictions discussed include the presence of negative exponents, fractional exponents, and radicals. Several examples are provided to demonstrate which expressions qualify as polynomials and which do not, focusing on the importance of integer exponents and the exclusion of variables in the denominator.
đ Leading Terms, Coefficients, and Polynomial Degree
The fourth paragraph explores how to identify the leading term, leading coefficient, and degree of a polynomial. Using several examples, the speaker shows how to extract these elements from different polynomial expressions. For instance, in Y = 4x^3 - 16x - 4 + x^4 - x^2, the leading term is x^4 and the leading coefficient is 1. Another example demonstrates expanding a factored form using the FOIL method to obtain the standard form of the polynomial.
Mindmap
Keywords
đĄPolynomial Function
đĄDegree
đĄLeading Coefficient
đĄConstant Term
đĄStandard Form
đĄCoefficients
đĄVariable
đĄExponent
đĄFactored Form
đĄQuadratic
đĄCubic
Highlights
Introduction to polynomial functions, including identifying degree, leading coefficient, and constant term.
Definition of a polynomial function: a function of the form p(x) = a_n * x^n + a_(n-1) * x^(n-1) + ... + a_0.
Explanation of the leading term and leading coefficient in polynomial functions.
Standard forms of polynomial functions using examples like p(x) = 2x^3 + 5x^2 + 7x - 5.
Arranging polynomial terms in decreasing order of exponents to represent them in standard form.
Discussion of common mistakes, such as including negative exponents or fractions in polynomial expressions.
Demonstrating how to rewrite polynomials into standard form, emphasizing the importance of exponent arrangement.
Explanation of variables and how they affect polynomial terms (e.g., no radical signs or denominators for variables in polynomial functions).
Detailed breakdown of converting expressions like f(x) = 4x^3 - 16x - 4 + x^4 into standard polynomial form.
Introducing terms like degree, leading term, and constant term through practical examples.
Real-life application of the FOIL method in expanding polynomial expressions.
Explanation of how to identify whether an expression is a polynomial based on restrictions like radical signs, fraction exponents, and negative exponents.
Illustration of different types of polynomials (e.g., quadratic, cubic) based on the degree of the highest exponent.
Analyzing factored forms and expanding them into standard forms using the FOIL method.
Final review of a complex polynomial expression and how to determine the leading term, leading coefficient, and degree.
Transcripts
[Music]
good day everyone so in this video
lesson we will discuss
about polynomial function especially
how to identify the degree leading
coefficient
constant term of polynomial function
okay what is polynomial functions
anuga by a polynomial function a
polynomial function is a function of the
form
p of x operating uh is let us p
of x in polynomial function at n is
equal to
a sub n times x raised to n plus
a sub n minus 1 times x raised to n
minus 1
plus a sub n minus 2 times x
raise a raise to n minus 2 plus
up to a sub 1 x plus a sub 0
and this one and your a sub n
is not equal to zero okay class so young
last lag is uh another
constant term so a big sub hand young
and nothing john is a non-negative
integer a sub zero
and a sub one up to a sub n are real
numbers
so it called coefficients and then
a sub n times x raised to
n so ito una class
is the leading term so when
standard form no polynomial function at
n
and taught nothing gen is leading term
and then
a sub n naught and it's a leading term
and tawagnathan is
leading coefficient and then
function so in this way so p of x
or predicting in other notation like f
of x
or y so p d ring and it'll class
or predict letter like for example g
of x p of uh m of
x so basta uh in other notation
predication all right in the long p of x
you might
have not n so like for example melon
tying p
p of x is equal to two x cubed plus five
x squared plus seven
x minus five better nothing is nanganito
f
of x is equal to two x cubed plus five
x squared plus seven x minus five or
predicting y equals two x cubed plus
five x squared
plus seven x minus five so my kita you
know a polynomial function
okay so another
2x cubed plus 5x squared plus 7x minus
5.
so that part you exponent
arrange into decreasing power so he
picks a b hand
so thing in lageta is a variable now i
am variable not indeed
x so that pattern exponent
arranged in decreasing orders a b sub
and then constant term so mean
polynomial function exponent
p of x is equal to 2 x cubed plus
seven x minus five okay on polynomial
function
so again
parama sulet nathan or my right in
standard form
that path
nothing is whole number long so that
pattern
negative exponent and then volume
fraction
x variable but nothing i want
fraction exponent and
radical sign and then
the patrolang variables
exponent in the fraction hindering
our variable radical sign
at well in the variable denominator
nothing
okay so write the given polynomial
function in standard form so sabi
so like for example we have f of x is
equal to 4
x cubed minus 16 x minus 4
plus x to the 4 power minus
x squared so panong hold it nothing uh
illegal going standard form your
polynomial function
exponent yeah arranged in decreasing
order okay arrange in decreasing order
so since is some variable on the menu
given uh
x variable x so thinking times a
variable
okay plus for x cube minus
x squared minus 16 x minus four
so it is standard form non-polynomial
function not in a given selector io
another we have y is equal to one
over six x to the fourth minus x squared
plus five
x to the fifth plus seven x cubed minus
five
so capacitors
so that is five x to the fifth power
plus one six x to the fourth power
plus seven x cubed minus x squared
minus 5 so
into standard form
3x times x that is 3x squared 3x times 1
that is
x five times x that is five x
five times one that is five and then it
co combination
similar terms
so copy three x squared three x plus
five x that is eight
x oh that patois equals zero eight
uh
of x is equal to 3x squared plus 8x plus
5.
okay another example so i determine not
encompolynomial function or not okay
basis
function for letter a okay my
restriction
a negative exponent of fractions
exponent
radical sign in variable my ah
my variable was a denominator voila so
therefore this is polynomial
letter b so letter b is not y
class time variables radical sign
polynomial number three a letter c
yes indirect polynomial bucket fraction
exponent
so not polynomial not a polynomial
for letter d y is equal to 1 minus 16
x squared polynomial by an or not
yes that is polynomial
restriction
radical sign a variable what are in
denominator
maritime variables are denominator
you know maritime variables a
denominator so hindito
polynomial how about this one
polynomial ba
okay yeah not a polynomial parenchy
okay proceed for example number i know
this one so pano natin ma determinion
leading term
leading coefficient and the
leading term latin is 2 x cube okay
exponent so the answer is 2 x cubed
and leading coefficient
okay so again nothing leading
coefficient that's a leading term
and then you degree pakistan
term ibis have been a constant constant
or mulasian casama
variable so alindito okay
so example that negative five
okay take notes of a sine huh negative
okay i'll give you more example
consider the given polynomial and fill
in the table below
okay halimbawayan so yeah in a factored
form so
standard yen so nothing standard
that is f of x is equal to 2x squared
plus 16x so
betting and not including your leading
term leading coefficient degrees
exponent that is two x squared and
leading coefficient that is two
okay thinking that then and young degree
and young pyramid as an exponent that is
two
and then this type is type of
polynomial function quadratic
so highest exponent is two and taught
net engine is quadratic
another f of x is equal to three x plus
five times
x plus one so now factored form que
laguna to expand
okay using foil method the answer is
three x squared plus eight x plus five
so capacity and leading term leading
coefficient
and degree so any leading term not n
that is three x squared
leading coefficient so the tuning
nothing cocooning
okay for example number three we have y
is equal to 4x cubed minus 16x minus 4
plus x to the fourth minus x squared so
so going out in standard form yeah so
that is
y is equal to x to the fourth plus four
x cubed minus
x squared minus sixteen x minus four
so on gagorina leading term and that is
x to the fourth
leading coefficient one since
leading coefficient is one okay
y is equal to a plus four times a
squared minus four eight plus sixteen
okay
during grade eight in the round so
sum and difference okay
so in expanding on let me multiply the
answer is
a cube plus 64 that is okay
a cube plus 64. so unknown leading
terminating that is a cube
leading coefficient one say well
so my one cn integral polynomial three
so capacity
and highest exponent and targeting is
cubic
okay number five y is equal to x squared
plus
x to the fourth minus three x to the
sixth plus x to the fifty plus ten
so indeed is in standard form
okay and then coordinating leading term
that is
negative three x to the sixth power
and then the leading coefficient is
negative three
and the degree of polynomial is six six
in pinah matas
so pagani toppak more than six now and
then that indeed
is uh starting
okay
thank you for watching this video i hope
you learned something
don't forget to like subscribe and hit
the bell button
put updated ko for more video tutorial
this is your guide in learning your mod
lesson your walmart channel
Voir Plus de Vidéos Connexes
[Tagalog] Write Polynomial Function into Standard Form, Determine the Degree, Leading Term, Constant
Polynomials - Classifying Monomials, Binomials & Trinomials - Degree & Leading Coefficient
Polinomial (Bagian 1) - Pengertian dan Operasi Aljabar Polinomial Matematika Peminatan Kelas XI
Adding Polynomials Horizontally
How to Divide Polynomials Using LONG DIVISION | Math 10
Synthetic Division of Polynomials
5.0 / 5 (0 votes)