3 mistakes learners make with the binomial expansion, each one named and explained.
Each term is a choose number times a power of the first part times a power of the second, and every part of a term carries its power.
The x squared term of (3 - 2x) to the 6 is given as 324, which is 3 to the 4 times 4 with no 15.
The powers are the visible part of the pattern, so the choose number in front is the easiest piece to drop.
Question. Find the coefficient of x squared in (3 - 2x) to the power 6.
A common answer. 324, using only the powers
The answer. 4860
Why. Every term is C(n, r) times the powers. Here that is C(6, 2) = 15, then 3 to the 4 times (-2) squared, giving 15 x 81 x 4 = 4860.
Write C(n, r) first, then the two powers. Three factors, every time.
An answer of -4860 where squaring the negative should have made it positive.
The minus is copied along out of habit, without checking what the power does to it.
Question. Is the x squared coefficient of (3 - 2x) to the 6 positive or negative?
A common answer. Negative, because the bracket has a minus in it
The answer. Positive
Why. The term carries (-2) squared, and a negative squared is positive. Odd powers keep the minus; even powers lose it.
Check whether the power on the negative part is odd or even before writing the sign.
(-2x) squared is treated as -2 times x squared instead of 4x squared.
The power sits next to the bracket and the eye attaches it to the letter it touches.
Question. What does (-2x) squared come to?
A common answer. -2x squared
The answer. 4x squared
Why. The bracket means the whole term is squared, so the -2 is squared as well: (-2) squared is 4.
A power on a bracket lands on everything inside it, numbers included.
WAJD spots these patterns in your child's answers and names the one behind their wrong answers, instead of just marking them wrong.