Introduction to Polynomial Multiplication
Polynomial multiplication is a fundamental algebraic operation that allows us to find the product of two or more polynomials. This skill is essential for solving various mathematical problems and has applications in many fields.
What You'll Learn
- Multiplying polynomials by monomials
- Using the FOIL method for binomials
- Special cases like squaring binomials and difference of squares
- Multiplying larger polynomials
- Real-world applications
Multiplying with a Monomial
When multiplying a polynomial by a monomial, we use the distributive property.
Distributive Property: $$a(b + c) = ab + ac$$
Example:
Simplify: $$5x - 2x[3x - (5x - 1)]$$
Step 1: Simplify inside the brackets
$$5x - 2x[3x - (5x - 1)]$$
$$= 5x - 2x[3x - 5x + 1]$$
$$= 5x - 2x[-2x + 1]$$
Step 2: Apply the distributive property
$$= 5x - 2x \cdot (-2x) + 2x \cdot 1$$
$$= 5x + 4x^2 + 2x$$
Step 3: Combine like terms
$$= 4x^2 + 7x$$
The FOIL Method
FOIL is an acronym that helps us remember how to multiply two binomials:
- First terms
- Outside terms
- Inside terms
- Last terms
Interactive FOIL Example: (2x-3)(5x+1)
Result: 10x² + 2x - 15x - 3
Simplified: 10x² - 13x - 3
Full Steps:
Multiply: $$(2x - 3)(5x + 1)$$
Using FOIL:
First: $$2x \cdot 5x = 10x^2$$
Outside: $$2x \cdot 1 = 2x$$
Inside: $$-3 \cdot 5x = -15x$$
Last: $$-3 \cdot 1 = -3$$
Combining all terms: $$10x^2 + 2x - 15x - 3$$
Simplified: $$10x^2 - 13x - 3$$
Special Cases
1. Squaring a Binomial
$$(a + b)^2 = a^2 + 2ab + b^2$$
$$(a - b)^2 = a^2 - 2ab + b^2$$
Example:
Simplify: $$(2x - 5)^2$$
Using the formula $$(a - b)^2 = a^2 - 2ab + b^2$$:
$$(2x - 5)^2 = (2x)^2 - 2(2x)(5) + 5^2$$
$$= 4x^2 - 20x + 25$$
2. Difference of Squares
$$(a + b)(a - b) = a^2 - b^2$$
Example:
Simplify: $$(2x - 3)(2x + 3)$$
Using the formula $$(a + b)(a - b) = a^2 - b^2$$:
$$(2x - 3)(2x + 3) = (2x)^2 - 3^2$$
$$= 4x^2 - 9$$
A. Multiplying with a Monomial
When multiplying a polynomial by a monomial, we use the distributive property to multiply each term of the polynomial by the monomial.
$$a(b + c) = ab + ac$$
Example:
Simplify: $$5x - 2x[3x - (5x - 1)]$$
Step 1: Simplify inside the brackets
$$5x - 2x[3x - (5x - 1)]$$
$$= 5x - 2x[3x - 5x + 1]$$
$$= 5x - 2x[-2x + 1]$$
Step 2: Apply the distributive property
$$= 5x - 2x \cdot (-2x) + 2x \cdot 1$$
$$= 5x + 4x^2 + 2x$$
Step 3: Combine like terms
$$= 4x^2 + 7x$$
B. Multiplying Binomials
There are several methods for multiplying binomials, including the FOIL method and using the distributive property.
Example:
Multiply: $$(2x - 3)(5x + 1)$$
Using FOIL:
First: $$2x \cdot 5x = 10x^2$$
Outside: $$2x \cdot 1 = 2x$$
Inside: $$-3 \cdot 5x = -15x$$
Last: $$-3 \cdot 1 = -3$$
Combining all terms: $$10x^2 + 2x - 15x - 3$$
Simplified: $$10x^2 - 13x - 3$$
Special Cases of Binomial Multiplication
1. Squaring a Binomial
$$(a + b)^2 = a^2 + 2ab + b^2$$
$$(a - b)^2 = a^2 - 2ab + b^2$$
Example:
Simplify: $$(2x - 5)^2$$
Using the formula $$(a - b)^2 = a^2 - 2ab + b^2$$:
$$(2x - 5)^2 = (2x)^2 - 2(2x)(5) + 5^2$$
$$= 4x^2 - 20x + 25$$
2. Difference of Squares
$$(a + b)(a - b) = a^2 - b^2$$
Example:
Simplify: $$(2x - 3)(2x + 3)$$
Using the formula $$(a + b)(a - b) = a^2 - b^2$$:
$$(2x - 3)(2x + 3) = (2x)^2 - 3^2$$
$$= 4x^2 - 9$$
C. Combination
Sometimes we need to combine multiple techniques to multiply more complex polynomial expressions.
Example:
Simplify: $$-5(3 - 4x)^2 + 3(5 - 2x)(5 + 2x)$$
Step 1: Simplify $$(3 - 4x)^2$$ using the square of a binomial formula $$(a - b)^2 = a^2 - 2ab + b^2$$
$$(3 - 4x)^2 = 3^2 - 2(3)(4x) + (4x)^2$$
$$= 9 - 24x + 16x^2$$
Step 2: Simplify $$(5 - 2x)(5 + 2x)$$ using the difference of squares formula $$(a - b)(a + b) = a^2 - b^2$$
$$(5 - 2x)(5 + 2x) = 5^2 - (2x)^2$$
$$= 25 - 4x^2$$
Step 3: Substitute back into the original expression
$$-5(9 - 24x + 16x^2) + 3(25 - 4x^2)$$
Step 4: Multiply through each parenthesis
$$-5(9 - 24x + 16x^2) = -45 + 120x - 80x^2$$
$$3(25 - 4x^2) = 75 - 12x^2$$
Step 5: Combine all terms
$$-45 + 120x - 80x^2 + 75 - 12x^2$$
$$= (-45 + 75) + 120x + (-80x^2 - 12x^2)$$
$$= 30 + 120x - 92x^2$$
Step 6: Rearrange into standard form (descending powers of x)
$$30 + 120x - 92x^2$$
$$= -92x^2 + 120x + 30$$
D. Multiplying Polynomials
For multiplying general polynomials, we multiply each term of one polynomial by each term of the other polynomial.
Example:
Multiply: $$(2x^2 - 3x + 4)(3x - 5)$$
Step 1: Multiply each term of the first polynomial by each term of the second
$$(2x^2)(3x) = 6x^3$$
$$(2x^2)(-5) = -10x^2$$
$$(-3x)(3x) = -9x^2$$
$$(-3x)(-5) = 15x$$
$$(4)(3x) = 12x$$
$$(4)(-5) = -20$$
Step 2: Combine like terms
$$6x^3 - 10x^2 - 9x^2 + 15x + 12x - 20$$
$$= 6x^3 - 19x^2 + 27x - 20$$
E. Application
Polynomial multiplication has many real-world applications, including calculating areas, volumes, and solving problems in physics and engineering.
Volume of a rectangular prism: $$Vol = l \times w \times h$$
Surface area of a rectangular prism: $$SA = 2lw + 2lh + 2wh$$
Example:
A rectangular prism has dimensions $$2x+3$$ by $$x-3$$ by $$x-4$$. Write a simplified expression for the surface area of the prism.
Step 1: Use the surface area formula
$$SA = 2(2x+3)(x-3) + 2(2x+3)(x-4) + 2(x-3)(x-4)$$
Step 2: Multiply each pair of binomials
$$(2x+3)(x-3) = 2x^2 - 6x + 3x - 9 = 2x^2 - 3x - 9$$
$$(2x+3)(x-4) = 2x^2 - 8x + 3x - 12 = 2x^2 - 5x - 12$$
$$(x-3)(x-4) = x^2 - 4x - 3x + 12 = x^2 - 7x + 12$$
Step 3: Substitute back into the surface area formula
$$SA = 2(2x^2 - 3x - 9) + 2(2x^2 - 5x - 12) + 2(x^2 - 7x + 12)$$
$$= 4x^2 - 6x - 18 + 4x^2 - 10x - 24 + 2x^2 - 14x + 24$$
$$= 4x^2 + 4x^2 + 2x^2 - 6x - 10x - 14x - 18 - 24 + 24$$
$$= 10x^2 - 30x - 18$$
Practice Problems
Test your understanding with these interactive practice problems:
Problem 1:
Multiply: $$(3x + 2)(x - 4)$$
Use the FOIL method:
F: 3x · x
O: 3x · (-4)
I: 2 · x
L: 2 · (-4)
Problem 2:
Simplify: $$(x + 5)^2$$
Use the formula for squaring a binomial:
(a + b)² = a² + 2ab + b²
Problem 3:
Multiply: $$(2x^2 - 3x + 4)(3x - 5)$$
Multiply each term of the first polynomial by each term of the second polynomial:
(2x²)(3x) + (2x²)(-5) + (-3x)(3x) + (-3x)(-5) + (4)(3x) + (4)(-5)
Select Problem Types to Generate:
Power Settings
This affects the maximum degree of polynomials in the "Higher Powers" category.