Wize High School Grade 10 Math Textbook > Expanding Polynomials

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

3x2\boxed{\Huge{3x^2}}3x2​
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^23x2 and 8x28x^28x2 are like terms
4x24x^24x2and 4x34x^34x3 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\ \ \ \ \ 37x2     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}4x+5y       5x2+4       −3t15−2t12
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+gx2+4x+3       −8g3−7g2+g

Practice: Terminology
Match the LETTER to the expression. You cannot repeat letters.

A.

coefficient

B.

monomial

C.

variable

D.

binomial

E.

trinomial

F.

constant

G.

exponent

3x+4x−53x+4x-53x+4x−5 in its simplified form

x2+3x−8x^2+3x-8x2+3x−8

the xxx in 2x32x^32x3

the 8 in 4d+84d+84d+8

3h53h^53h5

the 222 in 2x32x^32x3

the 333 in 2x32x^32x3

I don't know

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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})}3x2×(−4x3)

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

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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+2x−1)

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

=−12x5+6x3−3x2=-12x^5+6x^3-3x^2=−12x5+6x3−3x2

Example 3
(5x−3)(−2x)(5x-3)(-2x)(5x−3)(−2x)

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

=−10x2+6x=-10x^2+6x=−10x2+6x

Example 4
−(3x2−2x−5)-(3x^2-2x-5)−(3x2−2x−5)

=−3x2+2x+5=-3x^2+2x+5=−3x2+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)

4(2x−3)\bct4(2x-3)4(2x−3)

=4(2x)+4(−3)=\bct4(2x)+\bct4(-3)=4(2x)+4(−3)

=8x−12=8x-12=8x−12

b) 5x(2−x+3x2)5x(2-x+3x^2)5x(2−x+3x2)

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

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

=10x−5x2+15x3=10x-5x^2+15x^3=10x−5x2+15x3

c) (3n2−4)(−2n)(3n^2-4)(-2n)(3n2−4)(−2n)

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

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

=−6n3+8n=-6n^3+8n=−6n3+8n

d) −(3x2−2xy−4y+4)-(3x^2-2xy-4y+4)−(3x2−2xy−4y+4)

=−3x2+2xy+4y−4=-3x^2+2xy+4y-4=−3x2+2xy+4y−4

Practice: Multiplying a Polynomial & Monomial
Expand and simplify −3x(4x2+2x−5)-3x(4x^2+2x-5)−3x(4x2+2x−5).

Answer

I don't know