Multiplying Polynomials

Using the distributive property and FOIL method

Multiplying a Monomial by a Polynomial

Use the distributive property: $a(b + c) = ab + ac$

Example: $3x(2x + 5) = 3x \cdot 2x + 3x \cdot 5 = 6x^2 + 15x$

To multiply two binomials, use FOIL:

Example: $(x + 3)(x + 5)$

Result: $x^2 + 5x + 3x + 15 = x^2 + 8x + 15$

Square of a Sum: $(a + b)^2 = a^2 + 2ab + b^2$

Square of a Difference: $(a - b)^2 = a^2 - 2ab + b^2$

Difference of Squares: $(a + b)(a - b) = a^2 - b^2$

Multiply: $2x(3x + 4)$

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Use the distributive property:

$2x(3x + 4)$ $= 2x \cdot 3x + 2x \cdot 4$ $= 6x^2 + 8x$

Answer: $6x^2 + 8x$

Multiply using FOIL: $(x + 4)(x - 2)$

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Use FOIL:

F: $x \cdot x = x^2$ O: $x \cdot (-2) = -2x$ I: $4 \cdot x = 4x$ L: $4 \cdot (-2) = -8$

Combine: $x^2 - 2x + 4x - 8 = x^2 + 2x - 8$

Answer: $x^2 + 2x - 8$

Expand: $(2x - 3)^2$

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Use the pattern $(a - b)^2 = a^2 - 2ab + b^2$

Here $a = 2x$ and $b = 3$ :

$(2x)^2 - 2(2x)(3) + (3)^2$ $= 4x^2 - 12x + 9$

Alternative - FOIL: $(2x - 3)(2x - 3)$

Result: $4x^2 - 12x + 9$

Answer: $4x^2 - 12x + 9$

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