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Question

Use Pascal’s triangle to expand the expression left parenthesis x plus 1 right parenthesis squared

The correct answer is: Thus, the expansion is:


    ANSWER:
    Hint:
    Pascal's Triangle is a method to know the binomial coefficients of terms of binomial expression (x + y)n, where n can be any positive integer and x,y are real numbers. Pascal Triangle is represented in a triangular form, it is kind of a number pattern in the form of a triangular arrangement.
    We are asked to find the expansion of the expression left parenthesis x plus 1 right parenthesis to the power of 5 using Pascal’s triangle.
    Step 1 of 2:
    The given expression is left parenthesis x plus 1 right parenthesis to the power of 5 . Here, the value of n=5. Thus, there would be 5+1=6 terms.
    Hence, we should use the sixth row of the Pascal’s triangle to write the corresponding expansion.
    Step 2 of 2:


    Analyzing the figure, we have:
    left parenthesis x plus 1 right parenthesis to the power of 5 equals 1 open parentheses x to the power of 5 close parentheses plus 5 x to the power of 4 left parenthesis 1 right parenthesis to the power of 1 plus 10 x cubed left parenthesis 1 right parenthesis squared plus 10 left parenthesis x right parenthesis squared left parenthesis 1 right parenthesis cubed plus 5 x left parenthesis 1 right parenthesis to the power of 4 plus left parenthesis 1 right parenthesis to the power of 5
    equals x to the power of 5 plus 5 x to the power of 4 plus 10 x cubed plus 10 x squared plus 5 x plus 1
    Thus, the expansion is:left parenthesis x plus 1 right parenthesis to the power of 5 equals x to the power of 5 plus 5 x to the power of 4 plus 10 x cubed plus 10 x squared plus 5 x plus 1
    Note:
    The answer can be found using the binomial expansion of left parenthesis x plus y right parenthesis to the power of n equals sum from k equals 0 to n of   n C subscript k x to the power of n minus k end exponent y to the power of k where n greater or equal than 0 .

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