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Wize High School Grade 11 Math Textbook > Operations with Polynomials

Review

Combining TransformationsPrevious Section
Adding and Subtracting PolynomialsNext Section
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Terminology Review

Terms

A term is a constant (number) or variable raised to some exponent.

3x2\boxed{\Huge{3x^2}}
  • The letter is the variable.
  • Any number in front of the variable is the coefficient.
  • The exponent (power) is the small number to the top right of the variable.
  • A number without a variable is a constant.

Like terms have the same variable and exponent.
  • 3x23x^2 and 8x28x^2 are like terms
  • 4x24x^2and 4x34x^3 are not like terms

Wize Tip
We can add or subtract like terms → just add or subtract the coefficient (number) part and keep the variable part the same!

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Polynomials

A polynomial is made up of one or more terms.

  • A monomial has one term.
  • Examples: 7x2     4x     −3y     37x^2\ \ \ \ \ 4x\ \ \ \ \ -3y\ \ \ \ \ 3
  • A binomial has two terms separated by an add or subtract sign.
  • Examples: 4x+5y       5x2+4       −3t15−2t124x+5y\ \ \ \ \ \ \ 5x^2+4\ \ \ \ \ \ \ -3t^{15}-2t^{12}
  • A trinomial has three terms separated by an add or subtract sign.
  • Examples: x2+4x+3       −8g3−7g2+gx^2+4x+3\ \ \ \ \ \ \ -8g^3-7g^2+g

Practice: Terminology

Match the LETTER to the expression. You cannot repeat letters.
A.
exponent
B.
monomial
C.
binomial
D.
coefficient
E.
trinomial
F.
constant
G.
variable
3x+4x−53x+4x-5 in its simplified form
x2+3x−8x^2+3x-8
the xx in 2x32x^3
the 8 in 4d+84d+8
3h53h^5
the 22 in 2x32x^3
the 33 in 2x32x^3
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Multiplying by a Monomial

Multiplying two Monomials

We can multiply any two monomials together (they don't have to be like terms!) → just multiply the coefficients (number part) and the variable parts separately!

Example 1
3x2×(−4x3){\colorTwo{3}\colorThree{x^2}\times(\colorTwo{-4}\colorThree{x^3})}
=(3×(−4))(x2×x3)\bm{=\colorTwo{(3\times(-4))}\colorThree{(x^2\times x^3)}}
=(−12)(x5)\bm{=\colorTwo{(-12)}\colorThree{(x^5)}}
=−12x5\bm{=-12x^5}

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Multiplying a Monomial and a Polynomial

When multiplying a monomial and a polynomial, we have to use the distributive property.

Example 2
3x2(−4x3+2x−1)3x^2(-4x^3+2x-1)

=(3x2)(−4x3)+(3x2)(2x)+(3x2)(−1)=(3x^2)(-4x^3)+(3x^2)(2x)+(3x^2)(-1)

=−12x5+6x3−3x2=-12x^5+6x^3-3x^2


Example 3
(5x−3)(−2x)(5x-3)(-2x)

=(5x)(−2x)+(−3)(−2x)=(5x)(-2x)+(-3)(-2x)

=−10x2+6x=-10x^2+6x


Example 4
−(3x2−2x−5)-(3x^2-2x-5)

=−3x2+2x+5=-3x^2+2x+5
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Example: Multiplying a Polnomial & Monomial

Expand and simplify the following.

a) 4(2x−3)4(2x-3)

4(2x−3)\bct4(2x-3)

=4(2x)+4(−3)=\bct4(2x)+\bct4(-3)

=8x−12=8x-12

b) 5x(2−x+3x2)5x(2-x+3x^2)

5x(2−x+3x2)\bct{5x}(2-x+3x^2)

=5x(2)+5x(−x)+5x(3x2)=\bct{5x}(2)+\bct{5x}(-x)+\bct{5x}(3x^2)

=10x−5x2+15x3=10x-5x^2+15x^3

c) (3n2−4)(−2n)(3n^2-4)(-2n)

=(3n2−4)(−2n)=(3n^2-4)(\bct{-2n})

=3n2(−2n)−4(−2n)=3n^2(\bct{-2n})-4(\bct{-2n})

=−6n3+8n=-6n^3+8n

d) −(3x2−2xy−4y+4)-(3x^2-2xy-4y+4)

=−3x2+2xy+4y−4=-3x^2+2xy+4y-4

Practice: Multiplying a Polynomial & Monomial

Expand and simplify −3x(4x2+2x−5)-3x(4x^2+2x-5).