Expand (a+3b)²: Converting to Multiplication and Addition Forms

Video Solution

Solution Steps

00:00 Express as a sum and product

00:10 A squared term is actually a multiplication of itself by itself

00:25 We'll use the short multiplication formulas to expand the brackets

00:35 In our exercise A is the X

00:42 and 3B is the Y

00:47 We'll substitute according to the formula and solve

00:57 And this is the solution to the question

Step-by-Step Solution

To solve this problem, we need to express $(a+3b)^2$ in two forms: as a multiplication of like terms and as an addition of polynomial terms.

Step 1: Express as a multiplication
The expression $(a+3b)^2$ can be rewritten as $(a+3b)(a+3b)$ . This indicates the binomial is multiplied by itself.
Step 2: Expand using the binomial theorem
We apply the formula for squaring a binomial:
$(x+y)^2 = x^2 + 2xy + y^2$ .
Here, let $x = a$ and $y = 3b$ . Applying the formula we get:
$(a+3b)^2 = a^2 + 2(a)(3b) + (3b)^2$ .
Step 3: Simplify
Perform the calculations for each term:
- $a^2$ : The square of $a$ .
- $2(a)(3b) = 6ab$ : Multiply the terms and the coefficients.
- $(3b)^2 = 9b^2$ : Square the coefficient and the variable.
Combining these, the expanded form is:
$a^2 + 6ab + 9b^2$ .

Thus, the expression as a multiplication is $(a+3b)(a+3b)$ , and as an addition, it is $a^2 + 6ab + 9b^2$ .

Therefore, the solution to the problem is $(a+3b)(a+3b)$ and $a^2+6ab+9b^2$ .