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Identifying the degree and leading coefficient of a polynomial

For the following polynomials, identify the degree, the leading term, and the leading coefficient.

  1. 3 + 2 x 2 4 x 3
  2. 5 t 5 2 t 3 + 7 t
  3. 6 p p 3 2
  1. The highest power of x is 3, so the degree is 3. The leading term is the term containing that degree, −4 x 3 . The leading coefficient is the coefficient of that term, −4.
  2. The highest power of t is 5 , so the degree is 5. The leading term is the term containing that degree, 5 t 5 . The leading coefficient is the coefficient of that term, 5.
  3. The highest power of p is 3 , so the degree is 3. The leading term is the term containing that degree, p 3 , The leading coefficient is the coefficient of that term, −1.
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Identify the degree, leading term, and leading coefficient of the polynomial 4 x 2 x 6 + 2 x 6.

The degree is 6, the leading term is x 6 , and the leading coefficient is −1.

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Adding and subtracting polynomials

We can add and subtract polynomials by combining like terms, which are terms that contain the same variables raised to the same exponents. For example, 5 x 2 and −2 x 2 are like terms, and can be added to get 3 x 2 , but 3 x and 3 x 2 are not like terms, and therefore cannot be added.

Given multiple polynomials, add or subtract them to simplify the expressions.

  1. Combine like terms.
  2. Simplify and write in standard form.

Adding polynomials

Find the sum.

( 12 x 2 + 9 x 21 ) + ( 4 x 3 + 8 x 2 5 x + 20 )

4 x 3 + ( 12 x 2 + 8 x 2 ) + ( 9 x 5 x ) + ( −21 + 20 )      Combine like terms . 4 x 3 + 20 x 2 + 4 x 1    Simplify .

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Find the sum.

( 2 x 3 + 5 x 2 x + 1 ) + ( 2 x 2 3 x 4 )

2 x 3 + 7 x 2 −4 x −3

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Subtracting polynomials

Find the difference.

( 7 x 4 x 2 + 6 x + 1 ) ( 5 x 3 2 x 2 + 3 x + 2 )

7 x 4 5 x 3 + ( x 2 + 2 x 2 ) + ( 6 x 3 x ) + ( 1 2 )    Combine like terms . 7 x 4 5 x 3 + x 2 + 3 x 1 Simplify .

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Find the difference.

( −7 x 3 7 x 2 + 6 x 2 ) ( 4 x 3 6 x 2 x + 7 )

−11 x 3 x 2 + 7 x −9

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Multiplying polynomials

Multiplying polynomials is a bit more challenging than adding and subtracting polynomials. We must use the distributive property to multiply each term in the first polynomial by each term in the second polynomial. We then combine like terms. We can also use a shortcut called the FOIL method when multiplying binomials. Certain special products follow patterns that we can memorize and use instead of multiplying the polynomials by hand each time. We will look at a variety of ways to multiply polynomials.

Multiplying polynomials using the distributive property

To multiply a number by a polynomial, we use the distributive property. The number must be distributed to each term of the polynomial. We can distribute the 2 in 2 ( x + 7 ) to obtain the equivalent expression 2 x + 14. When multiplying polynomials, the distributive property allows us to multiply each term of the first polynomial by each term of the second. We then add the products together and combine like terms to simplify.

Given the multiplication of two polynomials, use the distributive property to simplify the expression.

  1. Multiply each term of the first polynomial by each term of the second.
  2. Combine like terms.
  3. Simplify.

Multiplying polynomials using the distributive property

Find the product.

( 2 x + 1 ) ( 3 x 2 x + 4 )

2 x ( 3 x 2 x + 4 ) + 1 ( 3 x 2 x + 4 )      Use the distributive property . ( 6 x 3 2 x 2 + 8 x ) + ( 3 x 2 x + 4 )    Multiply . 6 x 3 + ( −2 x 2 + 3 x 2 ) + ( 8 x x ) + 4    Combine like terms . 6 x 3 + x 2 + 7 x + 4      Simplify .

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Questions & Answers

if theta =30degree so COS2 theta = 1- 10 square theta upon 1 + tan squared theta
Martin Reply
how to compute this 1. g(1-x) 2. f(x-2) 3. g (-x-/5) 4. f (x)- g (x)
Yanah Reply
hi
John
hi
Grace
what sup friend
John
not much For functions, there are two conditions for a function to be the inverse function:   1--- g(f(x)) = x for all x in the domain of f     2---f(g(x)) = x for all x in the domain of g Notice in both cases you will get back to the  element that you started with, namely, x.
Grace
sin theta=3/4.prove that sec square theta barabar 1 + tan square theta by cosec square theta minus cos square theta
Umesh Reply
acha se dhek ke bata sin theta ke value
Ajay
sin theta ke ja gha sin square theta hoga
Ajay
I want to know trigonometry but I can't understand it anyone who can help
Siyabonga Reply
Yh
Idowu
which part of trig?
Nyemba
functions
Siyabonga
trigonometry
Ganapathi
differentiation doubhts
Ganapathi
hi
Ganapathi
hello
Brittany
Prove that 4sin50-3tan 50=1
Sudip Reply
f(x)= 1 x    f(x)=1x  is shifted down 4 units and to the right 3 units.
Sebit Reply
f (x) = −3x + 5 and g (x) = x − 5 /−3
Sebit
what are real numbers
Marty Reply
I want to know partial fraction Decomposition.
Adama Reply
classes of function in mathematics
Yazidu Reply
divide y2_8y2+5y2/y2
Sumanth Reply
wish i knew calculus to understand what's going on 🙂
Dashawn Reply
@dashawn ... in simple terms, a derivative is the tangent line of the function. which gives the rate of change at that instant. to calculate. given f(x)==ax^n. then f'(x)=n*ax^n-1 . hope that help.
Christopher
thanks bro
Dashawn
maybe when i start calculus in a few months i won't be that lost 😎
Dashawn
what's the derivative of 4x^6
Axmed Reply
24x^5
James
10x
Axmed
24X^5
Taieb
Thanks for this helpfull app
Axmed Reply
secA+tanA=2√5,sinA=?
richa Reply
tan2a+tan2a=√3
Rahulkumar
classes of function
Yazidu
if sinx°=sin@, then @ is - ?
NAVJIT Reply
the value of tan15°•tan20°•tan70°•tan75° -
NAVJIT

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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