Factor 30𝑎𝑏 squared plus 45𝑎 squared 𝑏 completely.
To factor or factorise an algebraic expression means to write it as a product rather than a sum. We’ll be applying the distributive property of multiplication here. If two terms have a common factor of 𝑥, then this factor can be taken out. 𝑥𝑦 plus 𝑥𝑧 can be written as 𝑥 multiplied by 𝑦 plus 𝑧.
To factor this expression completely means we need to find the greatest common factor of the two terms. To do this, we’ll first draw factor trees for each of the two terms.
Let’s start with 30𝑎𝑏 squared. This can be written as the product of 30 multiplied by 𝑎𝑏 squared. Both of these can be broken down further. So we’ll continue each branch of the factor tree. 30 can be written as a product of 10 and three. Three is a prime number. So this branch of the factor tree stops here. 10, however, can be written as the product of two and five, both of which are prime numbers.
Now let’s consider 𝑎𝑏 squared. This can be split into the product of 𝑎 and 𝑏 squared. 𝑏 squared can be further split into the product of 𝑏 and 𝑏. Collecting out all of these factors, we can write 30𝑎𝑏 squared as two multiplied by three multiplied by five multiplied by 𝑎 multiplied by 𝑏 multiplied by 𝑏.
Now let’s consider 45𝑎 squared 𝑏. This can be written as the product of 45 and 𝑎 squared 𝑏. 45 can be expressed as the product of nine and five. Five is a prime number. But nine is not. Nine is the product of three and three, which are prime.
𝑎 squared 𝑏 can be written as the product of 𝑎 squared and 𝑏. And 𝑎 squared can be broken down further into 𝑎 and 𝑎. Collecting out all of these factors, we can express 45𝑎 squared 𝑏 as three multiplied by three multiplied by five multiplied by 𝑎 multiplied by 𝑎 multiplied by 𝑏.
Now let’s determine the greatest common factor of these two terms. Looking through the list of factors, we can see that they both share a factor of three. They also share a factor of five, one factor of 𝑎, and one factor of 𝑏. Their greatest common factor is, therefore, the product of these shared factors, three multiplied by five multiplied by 𝑎 multiplied by 𝑏, which is 15𝑎𝑏. So the greatest common factor of these two terms, which we write outside the bracket, is 15𝑎𝑏.
Now I need to determine what goes inside the bracket. There’s an addition sign in the bracket, as the sign in the original expression was plus. We now just need to collect all of the leftover factors that weren’t included in the greatest common factor.
For 30𝑎𝑏 squared, that’s a factor of two and a factor of 𝑏. The first term in the bracket is, therefore, two 𝑏. For 45𝑎 squared 𝑏, the leftover terms are three and 𝑎. The second term in the bracket is, therefore, the product of these, three 𝑎.
We’ve now factored the original expression completely as it’s written as a product. And the two terms in the bracket don’t share any common factors. The factor format is 15𝑎𝑏 multiplied by two 𝑏 plus three 𝑎.