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Expand the following: (a) (a+b)(a-b) (b) (2x+3y)(4x-y)

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4 parts
a

a2−b2a^2 - b^2

b

8x2+10xy−3y28x^2 + 10xy - 3y^2

c

4m2+4mn+n24m^2 + 4mn + n^2

d

5x2−6xy+y25x^2 - 6xy + y^2

Step-by-step solution

9 steps

Let's expand and simplify each expression from Example 4.

Part a· 2 steps

Expand (a+b)(a−b)(a+b)(a-b)

  1. Apply the distributive property (FOIL method).

    a(a)+a(−b)+b(a)+b(−b)a(a) + a(-b) + b(a) + b(-b) a2−ab+ab−b2a^2 - ab + ab - b^2

  2. Combine like terms.

    a2+(−ab+ab)−b2a^2 + (-ab + ab) - b^2 a2+0−b2a^2 + 0 - b^2 a2−b2\boxed{a^2 - b^2}

Part b

8x2+10xy−3y28x^2 + 10xy - 3y^2

Expand (2x+3y)(4x−y)(2x+3y)(4x-y)

  1. Apply the distributive property (FOIL method).

    2x(4x)+2x(−y)+3y(4x)+3y(−y)2x(4x) + 2x(-y) + 3y(4x) + 3y(-y) 8x2−2xy+12xy−3y28x^2 - 2xy + 12xy - 3y^2

  2. Combine like terms.

    8x2+(−2xy+12xy)−3y28x^2 + (-2xy + 12xy) - 3y^2 8x2+10xy−3y28x^2 + 10xy - 3y^2 8x2+10xy−3y2\boxed{8x^2 + 10xy - 3y^2}

Part c

4m2+4mn+n24m^2 + 4mn + n^2

Expand (2m+n)2(2m+n)^2

  1. Rewrite the expression as a product of two binomials.

    (2m+n)(2m+n)(2m+n)(2m+n)

  2. Apply the distributive property (FOIL method).

    2m(2m)+2m(n)+n(2m)+n(n)2m(2m) + 2m(n) + n(2m) + n(n) 4m2+2mn+2mn+n24m^2 + 2mn + 2mn + n^2

  3. Combine like terms.

    4m2+(2mn+2mn)+n24m^2 + (2mn + 2mn) + n^2 4m2+4mn+n24m^2 + 4mn + n^2 4m2+4mn+n2\boxed{4m^2 + 4mn + n^2}

Part d

5x2−6xy+y25x^2 - 6xy + y^2

Expand (5x−y)(x−y)(5x-y)(x-y)

(Assuming the missing term is yy based on common algebraic patterns.)

  1. Apply the distributive property (FOIL method).

    5x(x)+5x(−y)+(−y)(x)+(−y)(−y)5x(x) + 5x(-y) + (-y)(x) + (-y)(-y) 5x2−5xy−xy+y25x^2 - 5xy - xy + y^2

  2. Combine like terms.

    5x2+(−5xy−xy)+y25x^2 + (-5xy - xy) + y^2 5x2−6xy+y25x^2 - 6xy + y^2 5x2−6xy+y2\boxed{5x^2 - 6xy + y^2}
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