# 10.3 – Multiplying Binomials

## Key Terms

• Constant Polynomials – Polynomials of degree 0.
• A constant polynomial has no variable.
• Distributive Property – The rule that if a, b, and c are numbers or expressions, then $a\bullet (b+c)=a\bullet b+a\bullet c$.
• FOIL – A method of multiplying binomials. FOIL stands for First, Outer, Inner, Last: the order in which you would multiply a binomial’s terms.
• Linear Polynomial – A polynomial of degree 1.
• A linear polynomial has a variable term and may also have a constant term.

## Review

Remember…
• The area of a rectangle is the number of square units it covers.

• Degree
• Don’t confuse the degree of a polynomial with the number of terms in the polynomial.
• The degree of a polynomial is its greatest exponent.
• x is a polynomial of degree 1 because $x=x^{1}$. It’s because there is only ONE x, not because x has only 1 term.
• ALWAYS write your answers in descending order (greatest to smallest degree).

• Tiles
• Notice that the tile tool assumes that all terms are separated by plus signs (+).
• Why? Because a polynomial is a sum of terms.

• Factors
• A factor is a number or expression that is multiplied.
• A product is the answer to a multiplication problem.
• The commutative property of multiplication applies to polynomials and their factors.
• The order of the factors does not change the product.

This can also be written as:
$(x)(x+1)=x^{2}+x$

## Notes

Polynomial Multiplication Using Tiles
• Using tiles helps you visualize the product of factors that have added items.
• We cannot use the tile tool to multiply polynomial factors that have subtracted terms.
• Steps to Multiply with Tiles
1. For the first factor, put two x-tiles and one 1-tile in the box above the blue region.
2. For the second factor, put one x-tile and two 1-tiles in the box to the left of the blue region.
3. When you are done, enter the product in the box at the bottom.
1. Enter the answer using the caret ( ^ ) for exponents.
2. Example, enter x^2 for $x^{2}$.
• Watch the video below to see what happens when we add tiles to the factor areas.

• Example: $(2x+1)(x+2)$

Answer: $2x^{2}+5x+2$

The Distributive Property
• Using tiles is helpful for multiplying constant polynomials and linear polynomials, but if you need to multiply higher-degree polynomials, you need to use the distributive property.
• You can find the product of ANY two binomials using the distributive property .
• In the distributive property, a, b, and c can be numbers, variables, entire variable expressions, or polynomials (which ARE variable expressions).
• Distribute even further…

• Another example

• More examples

FOIL – First, Outer, Inner, Last
• Multiplying Binomials Using FOIL

• Example

FOIL – Special Cases

## Examples

 Ex 1. What are the factors of the product represented below? Ex 2. Which polynomial is the product of (3x + 2) and (x + 4)? Ex 3. The area of Natalie’s garden after she adds a border x feet wide is equal to (3 + 2x)(6 + 2x). Which polynomial represents the area of Natalie’s garden? Ex 4. What is the product of the polynomials? $(4x^{3}+8x)(x^{2}-3)$ Use FOIL: $4x^{5}-12x^{3}+8x^{3}-24x$ Combine like terms for the answer: $4x^{5}-4x^{3}-24x$ Ex 5. What is the product of the polynomials below? $(8-x^{2})(x-x^{3})$ FOIL: $8x-8x^{3}-x^{3}+x^{5}$ Reorder to descending order and combine like terms for the answer: $x^{5}-9x^{3}+8x$ Ex 6. Use the FOIL method to find the product of the binomials. $(x^{2}+7)(x^{2}+7)$ FOIL (answer): $x^{4}+14x^{2}+49$