Expansion takes an expression from factored form and uses the distributive property to remove the brackets and write the expression as a sum of products. The distributive property is:
$A\left(B+C\right)=AB+AC$A(B+C)=AB+AC
The distributive property can be used to simplify or expand an expression with a linear factor (degree equals one) multiplied by a constant. For example, $2\left(b+3\right)=2b+6$2(b+3)=2b+6.
The distributive property can be used over more terms:
$A\left(B+C+D\right)=AB+AC+AD$A(B+C+D)=AB+AC+AD
Or applied multiple times, known as binomial expansion:
| $\left(A+B\right)\left(C+D\right)$(A+B)(C+D) | $=$= | $A\left(C+D\right)+B\left(C+D\right)$A(C+D)+B(C+D) |
| $=$= | $AC+AD+BC+BD$AC+AD+BC+BD |
The process of binomial expansion can be simplified by remembering that the first term multiplies each in the second bracket, and the second term multiplies each in the second bracket.
Hence, $\left(x+5\right)\left(x+2\right)=x^2+2x+5x+10$(x+5)(x+2)=x2+2x+5x+10. The like terms $2x$2x and $5x$5x can then be added together to produce the trinomial $x^2+7x+10$x2+7x+10.
After expanding, check if the expression can be simplified by collecting like terms.
Practice questions
Question 1
Expand and simplify the following:
$\left(2n+5\right)\left(5n+2\right)-4$(2n+5)(5n+2)−4
Question 2
Expand and simplify the following expression $\left(x+7\right)\left(x-7\right)-\left(x-3\right)^2$(x+7)(x−7)−(x−3)2
Further expansion
When multiplying more than two expressions, the easiest way to find the answer is to only multiply any two expressions at a time. Alternatively, a rule if known could be used to simplify the expansion.
For example, $\left(A+B\right)^3=\left(A+B\right)\left(A+B\right)^2$(A+B)3=(A+B)(A+B)2. Using this fact, an expression like $\left(x+6\right)^3$(x+6)3 can be expanded by first expanding $(x+6)^2=(x+6)(x+6)=x^2+12x+36$(x+6)2=(x+6)(x+6)=x2+12x+36 then multiplying this answer by $x+6$x+6 to find the fully expanded polynomial that equals $\left(x+6\right)^3$(x+6)3.
Practice questions
Question 3
Expand the following:
$\left(2x+1\right)\left(5x+7\right)\left(2x-1\right)$(2x+1)(5x+7)(2x−1)
Question 4
Expand $\left(3c+2\right)\left(2c^2+2\right)^2$(3c+2)(2c2+2)2.
Finding the degree and coefficients by expansion
Sometimes, only one part of a polynomial expansion needs to be identified. This could be the coefficient of a variable, the power of a variable or the degree of the polynomial. This can be done without having to expand the whole expression, by only focusing on certain parts of the polynomial.
To find the coefficient of any variable or power of a variable, identify where these terms will appear when expanding. Then, only expand those terms, and simplify the answer.
To find the degree of a single variable polynomial, this must be equal to the highest power of that variable when the polynomial is expanded. When the factors of a polynomial are expanded, powers of the variable will be added together. The highest power in the polynomial will be the combination of the highest powers from each factor. This means that the degree of the polynomial will be equal to the sum of the degree of each factor. It might be helpful to write any repeated factors out in full when calculating the degree for each factor.
Worked examples
example 1
Without fully expanding, find the coefficient of $x^2$x2 and determine the degree of the polynomial $\left(x^2+3x-1\right)\left(2x^3-6x^2-4x+8\right)$(x2+3x−1)(2x3−6x2−4x+8)
Think: The degree of the polynomial equals the sum of the degree of each factor. The degree of $x^2+3x-1$x2+3x−1 is $2$2, while the degree of $2x^3-6x^2-4x+8$2x3−6x2−4x+8 is $3$3. The polynomial expression when expanded will have degree $2+3=5$2+3=5.
Do: To find the coefficient of $x^2$x2, look for any terms which, when multiplied together, will result in an $x^2$x2 term. These are shown below:
Expanding these terms only and simplifying, we have $8x^2-12x^2+6x^2=2x^2$8x2−12x2+6x2=2x2. We can see without fully expanding the polynomial that the coefficient of $x^2$x2 is $2$2.
Practice questions
QUESTION 5
By using the fact that $A^3=AA^2$A3=AA2, expand and simplify $\left(x+3\right)^3$(x+3)3.
QUESTION 6
Without simplifying the whole expression, find just the coefficient of $x$x in the polynomial $\left(x^2-3x-5\right)\left(5x-4\right)$(x2−3x−5)(5x−4).
Applications of expansion
Developing polynomial expressions for a given situation is a very useful skill when working with geometric problems. It is usually a good idea to draw a diagram to represent the situation and label it with any known information and use variables to represent any unknowns. The problem can then be written as an algebraic expression or equation and solved using methods such as expansion and simplification.
Practice questions
QUESTION 7
A piece of wire $90$90 cm long is bent into the shape of a pentagon as shown.
Form an expression for $y$y in terms of $x$x.
Form an expression for the area of the pentagon in terms of $x$x. Leave your answer in expanded form.