Question Video: Factorizing Trinomials by Taking Out the Greatest Common Factor Mathematics

Factorize fully 48𝑚⁴ + 48𝑚²𝑛 − 15𝑛².

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Video Transcript

Factorize fully 48𝑚 to the fourth power plus 48𝑚 squared 𝑛 minus 15𝑛 squared.

We begin by observing that the coefficients of all three terms are multiples of three. Hence, we can factor the entire trinomial by three, which gives us three multiplied by 16𝑚 to the fourth power plus 16𝑚 squared 𝑛 minus five 𝑛 squared. Now consider the structure of the trinomial. The first term involves 𝑚 to the fourth power, which is equal to 𝑚 squared all squared. And the third term involves 𝑛 squared. The central term involves a product of 𝑚 squared and 𝑛. This suggests that the factored form of the trinomial is 𝐴𝑚 squared plus 𝐵𝑛 multiplied by 𝐶𝑚 squared plus 𝐷𝑛, for values of 𝐴, 𝐵, 𝐶, and 𝐷 to be determined.

We now need to find two numbers whose sum is equal to the coefficient of 𝑚 squared 𝑛, in this case 16, and whose product is equal to the product of the coefficients of the first and last terms. 16 multiplied by negative five is equal to negative 80. The factor pairs of 80 are as shown. As the product should be negative 80, we require a factor pair with opposite signs such that their sum is equal to 16. The correct pair are 20 and negative four. Rewriting the second term in the trinomial as the sum of two terms with these coefficients gives 16𝑚 to the fourth power plus 20𝑚 squared 𝑛 minus four 𝑚 squared 𝑛 minus five 𝑛 squared.

Separating this four-term expression into two binomials and factoring each separately gives four 𝑚 squared multiplied by four 𝑚 squared plus five 𝑛 minus 𝑛 multiplied by four 𝑚 squared plus five 𝑛. Hence, the fully factored form of the trinomial is three multiplied by four 𝑚 squared minus 𝑛 multiplied by four 𝑚 squared plus five 𝑛.