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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

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
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The sequence is {1,-1,1-1.....} has
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circular region of radious
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Sin(A+B) = sinBcosA+cosBsinA
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Prove it
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Joel
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
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7.5 and 37.5
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find the sum of 28th term of the AP 3+10+17+---------
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I think you should say "28 terms" instead of "28th term"
Vedant
the 28th term is 175
Nando
192
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if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
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write down the polynomial function with root 1/3,2,-3 with solution
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if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
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write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
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Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
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In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
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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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