Luku 2.1 (9th Grade Mathematics)

What is a polynomial?

  • Polynomial
  • The degree of a polynomial
  • Arranging polynomials
  • The value of a polynomial
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Polynomial

A polynomial P is an expression we get from one or more variables, constants and exponents of variables through addition, subtraction or multiplication.

A polynomial can have several different variables.

  • A polynomial with one variable x is marked as P(x).
  • A polynomial with two variables is marked as P(x, y).
  • A single number is called a constant term.

Note

  • A polynomial that contains only one term is called a monomial.
  • A polynomial that contains two terms is called a binomial.
  • A polynomial that contains three terms is called a trinomial.

Example 1

Monomial

  1. A monomial with one variable:
    P\left(x\right)=9x^7
  2. A monomial with two variables:
    ​​​M\left(a,\ b\right)=2ab

Binomial

  1. A binomial with one variable:
    Q\left(x\right)=3x^4-x
  2. A binomial with two variables:
    ​​B\left(x,\ y\right)=6x^3-2y^2​

Trinomial

  1. A trinomial with one variable:
    ​​T\left(x\right)=4x^6-3x^4+1​
  2. A trinomial with three variables:
    ​S\left(l,\ m,\ n\right)=l^2+2m+5n​
        • y2-y
        • -2ab
        • 3x2y
        • ab-2
        • x2-y+1
        • x2+y+3
        • a+b-2
        • 3x2+y
        • –3
        • –2
        • –1
        • 0
        • 1
        • 2
        • 3
        • 4
        • 5

        Polynomial

        Degree

        Constant term

        5x^4-x^2-3

        -2y^3+y^5-1

        3x^3-x^2-x

        -y^2+3y+3

        1+2x+3x^3

        The degree of a polynomial

        The exponent of a term is called the degree of the term. The exponent of the term with the greatest degree is the degree of the polynomial.

        Example 2

        In the polynomial P\left(x\right)=5x^3+2x^2-x+7 

        • ​the degree is 3, because the term with the greatest exponent is 5x3,
        • the degree of the constant term is zero, because 

        7\cdot x^0=7\cdot1=7.

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        The value of a polynomial

        Example 3

        The value of a polynomial can be calculated when the value of the variable is known.

        Let's calculate the value of the polynomial Q\left(x\right)=2x^3-x^2+3x-6 when x = 3.

        The marking Q(3) means that the value of the polynomial Q can be calculated when the variable x is replaced with the number 3.

        Q\left(3\right)=2\cdot3^3-3^2+3\cdot3-6=48

        Arranging a polynomial

        The value of the polynomial does not change when the order of the terms changes.

        In general, the terms of a polynomial are arranged in order of degree, beginning with the term with the greatest degree. When doing so, the constant term is always given last.

        Q\left(x\right)=2x^3-x^2+3x-6

        P(x) =

        • -2x7
        • +5
        • +3x4
        • +5x6
        • -2x
        • -5x2

        Q(y) =

        • +4y5
        • +y2
        • +2y4
        • -7y3
        • +y
        1. R\left(x\right)=2x^3-5x^2+1
          R\left(2\right)=
          2 ⋅ 3 – 5 ⋅ 2 + 1 = 
        2. T\left(y\right)=-3y^4+y^3
          T\left(-1\right)=
          –3 ⋅ ()4 + ()3 = 
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        Practise and solve

        Polynomial

        Degree

        Constant term

        Type

        3x^3+1

        -3x

        5x^2-2

        -x^5+x-1

        -12x^6

        1. P(0) = 
        2. P(2) = 
        3. P(–2) = 
        1. P(0) = 
        2. P(2) = 
        3. P(–2) =  
        1. Q(0) = 
        2. Q(–1) = 
        3. Q(3) = 
        1. R(1) = 
        2. R(–1) = 
        3. R(2) = 
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        Good to know

        P\left(x\right)=5x-3,Q\left(x\right)=3x+5,a=7 and b=-18.

        1. If P\left(x\right)=a, then x = .
        2. If P\left(x\right)=b, then x = .
        3. If P\left(x\right)=Q\left(x\right), then x = .

        P\left(x\right)=x^2-1,Q\left(x\right)=x^2+2x+3, m=-1 and n=3.

        1. If P\left(x\right)=m,then x = .
        2. IfP\left(x\right)=Q\left(x\right), then x = .
        3. If P\left(x\right)=n,then x = 
        • –2
        • –1
        • 0
        • 1
        • 2
        • 3

        Values of polynomials

        • 2+x3
        • x3-6x
        • 3x-2x2+1
        • x2+2x-3
        • 2+x3
        • 3x-2x2+1
        • x2+2x-3
        • x3-6x
        • 3x-2x2+1
        • x2+2x-3
        • 2+x3
        • x3-6x
        • If x = 3 cm, then 
          p =  cm.
        • If the perimeter is 48 cm, then
          x =  cm.
        • If x = 5 cm, then
          p =  cm.
        • If the perimeter is 56 cm, then
          x =  cm.
        • If x = 4 cm, then
          p =  cm.
        • If p = 72 cm, then
          x =  cm.
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        Extra

        The values of a polynomial

        The graph presents all the possible values of the polynomial Q(x) = x4 – 2x3 – 3, when the value of the variable x is between the integers –1 and 2.

        • –4
        • –3
        • – 2
        • –1
        • 0
        • 1
        • 2
        • 1.5
        • 0.5
        • –0.5
        • –1.5
        1. If x = –1, then Q(x) = 
        2. If x = 1, then Q(x) = 
        3. If x = , then Q(x) = –2.6875
        4. If x = , then Q(x) = –3.1875
        • –4
        • –3
        • – 2
        • –1
        • 0
        • 1
        • 2
        • 1.5
        • 0.5
        • –0.5
        • –1.5
        • =
        • <
        • >
        1. The value of the polynomial is the smallest when
           x = 
        2. Compare: Q(0)  Q(2)
        3. Compare: Q(0.8)  Q(1.9)
        4. Compare: Q(1.3)  Q(1.8)
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