Expanding and Simplifying

Every expansion in this unit comes from one rule, the distributive property: $a(b+c)=ab+ac$. The number outside the bracket multiplies every term inside. With a monomial in front, multiply the coefficients and add the exponents of matching variables: $3x(2x-5)=3x\cdot 2x+3x\cdot (-5)=6x^2-15x\text{.}$ A binomial times a binomial is the same rule used twice. Treat the first bracket as two separate terms and distribute each one over the second bracket:

Count the products: each of the 2 terms in the first bracket meets each of the 2 terms in the second, so there are $2\times 2=4$ products before you simplify. Some people remember the four products with the word FOIL: First ($x\cdot x$), Outer ($x\cdot 3$), Inner ($5\cdot x$), Last ($5\cdot 3$). FOIL is only a checklist for binomial times binomial. The distributive property is the real reason it works, and it also handles longer brackets, where FOIL has no letters left.

The area model makes the four products visible. Split each side of a rectangle into its terms, draw the grid lines, and write the area of each piece in its cell. Then add the cells. The grid also works when a term is negative or when there are two variables: you simply write signed products in the cells. For $(2x-y)(x+3y)$ the four cells are $2x\cdot x=2x^2$, $2x\cdot 3y=6xy$, $-y\cdot x=-xy$ and $-y\cdot 3y=-3y^2$. The two middle cells are like terms, so $6xy-xy=5xy$ and the product is $2x^2+5xy-3y^2$.

Multiplying binomials: area model | Mathematics II | High School Math | Khan Academy

The exponent in ${(2x+5)}^2$ means the bracket is multiplied by itself: $(2x+5)(2x+5)$. Expanding gives $4x^2+10x+10x+25=4x^2+20x+25\text{.}$ The two middle products are identical, which is why the middle term is twice the product of the two terms. In general, ${(a+b)}^2=a^2+2ab+b^2$ and ${(a-b)}^2=a^2-2ab+b^2$. One of the most common errors in this unit is writing ${(2x+5)}^2=4x^2+25$. Test it with $x=1$: the bracket gives $7^2=49$, but $4+25=29$. The missing $20x$ is the two middle rectangles of the area model.

Expand a few products of the form $(x+m)(x+n)$ and look for a pattern: $(x+1)(x+2)=x^2+3x+2$, $(x+2)(x+3)=x^2+5x+6$, $(x+3)(x+4)=x^2+7x+12$. The coefficient of $x$ is the sum of the two numbers and the constant is their product: $(x+m)(x+n)=x^2+(m+n)x+mn$. So $(x+5)(x+6)=x^2+11x+30$ can be written in one step. Run this pattern backwards and you have the method for factoring trinomials, two lessons from now.

To simplify an expression that contains several products, expand each product inside its own brackets first, then deal with what is outside. In $2(x+3)(x-1)-{(x-4)}^2$, expand $(x+3)(x-1)=x^2+2x-3$ and ${(x-4)}^2=x^2-8x+16$. Now multiply the first by 2 and subtract the whole second bracket: $2x^2+4x-6-x^2+8x-16=x^2+12x-22\text{.}$ Keeping the brackets until the last moment stops the subtraction from reaching only the first term. Computer algebra systems (CAS) and algebra tiles can confirm an expansion, and you can always check by hand: substitute a small number such as $x=2$ into the original and into your answer. For $(2x+3)(x-5)$ the original gives $7\times (-3)=-21$ and the expansion $2x^2-7x-15$ gives $8-14-15=-21$. If the two values differ, there is an error. If they agree, the answer is very likely right; test a second value to be more certain.