Find the sum of the coefficients of the first three terms that result from the expansion of 𝑥 minus two all raised to the fourth power according to the descending powers of 𝑥.
In this question, we are given a binomial expression in the form 𝑎 plus 𝑏 all raised to the 𝑛th power, where the value of our exponent or power 𝑛 is four. We will begin by recalling how we can expand an expression of this type using Pascal’s triangle and the binomial theorem.
Pascal’s triangle is a triangular array of the binomial coefficients. In this question, we were interested in the row 𝑛 equals four, which has five elements. The first term of the expansion of 𝑎 plus 𝑏 all raised to the fourth power is one multiplied by 𝑎 to the fourth power. The second term is equal to four multiplied by 𝑎 cubed multiplied by 𝑏 or 𝑏 to the first power. The third term is equal to six multiplied by 𝑎 squared multiplied by 𝑏 squared. Noticing that the powers or exponents of 𝑎 are decreasing and the exponents or powers of 𝑏 are increasing, our last two terms are four multiplied by 𝑎 multiplied by 𝑏 cubed and one multiplied by 𝑏 to the fourth power.
In this question, we want to expand 𝑥 minus two all raised to the fourth power. This means that the value of 𝑎 is 𝑥 and the value of 𝑏 is negative two. Substituting these values into our expression, we get one multiplied by 𝑥 to the fourth power plus four 𝑥 cubed multiplied by negative two plus six 𝑥 squared multiplied by negative two squared plus four 𝑥 multiplied by negative two cubed plus one multiplied by negative two to the fourth power. This in turn simplifies to one 𝑥 to the fourth power minus eight 𝑥 cubed plus 24𝑥 squared minus 32𝑥 plus 16.
Our terms have now been written according to the descending powers of 𝑥. We are interested in the coefficients of the first three terms. These are equal to one, negative eight, and 24. We are asked to find the sum of these three values. One plus negative eight is equal to negative seven. And adding 24 to this gives us 17. The sum of the coefficients of the first three terms of the expansion is 17.