Question
Use Pascal triangle to expand (x + y)6
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 binomial expansion of 6 (x + y)6 using Pascal’s formula.
The correct answer is: 6Cr
Step 1 of 1:
The Pascal’s triangle is given by:
![](data:image/png;base64,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)
From the triangle, the seventh line would give the coefficients of the expansion of the polynomial ( x + y)6 . Thus, we have:
.
The expansion of the polynomial (x + y)6 can be found using the values of 6Cr
Related Questions to study
Use polynomial identities to multiply the expressions ?
![open parentheses 25 straight x squared minus 15 straight y squared close parentheses open parentheses 25 straight x squared plus 15 straight y squared close parentheses](data:image/png;base64,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)
Use polynomial identities to multiply the expressions ?
![open parentheses 25 straight x squared minus 15 straight y squared close parentheses open parentheses 25 straight x squared plus 15 straight y squared close parentheses](data:image/png;base64,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)
How can you use polynomial identities to factor polynomials and simplify numerical expressions ?
![12 cubed plus 2 cubed](data:image/png;base64,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)
How can you use polynomial identities to factor polynomials and simplify numerical expressions ?
![12 cubed plus 2 cubed](data:image/png;base64,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)
Use Pascal triangle to expand ![left parenthesis x plus y right parenthesis to the power of 5](data:image/png;base64,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)
The answer cam also be found by expanding the formula of 5Cr .
Use Pascal triangle to expand ![left parenthesis x plus y right parenthesis to the power of 5](data:image/png;base64,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)
The answer cam also be found by expanding the formula of 5Cr .
Use polynomial identities to multiply the expressions ?
![open parentheses 3 x squared minus 4 x y close parentheses open parentheses 3 x squared plus 4 x y close parentheses](data:image/png;base64,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)
Use polynomial identities to multiply the expressions ?
![open parentheses 3 x squared minus 4 x y close parentheses open parentheses 3 x squared plus 4 x y close parentheses](data:image/png;base64,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)
How can you use polynomial identities to factor polynomials and simplify numerical expressions ?
![27 x to the power of 9 minus 343 y to the power of 6](data:image/png;base64,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)
How can you use polynomial identities to factor polynomials and simplify numerical expressions ?
![27 x to the power of 9 minus 343 y to the power of 6](data:image/png;base64,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)
Use polynomial identities to multiply the expressions ?
![left parenthesis 10 plus 21 right parenthesis squared](data:image/png;base64,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)
Use polynomial identities to multiply the expressions ?
![left parenthesis 10 plus 21 right parenthesis squared](data:image/png;base64,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)
How is ( x + y )n obtained from (x + y)n-1
You could also get the value of from
by just multiplying a ( x + y) with
.
How is ( x + y )n obtained from (x + y)n-1
You could also get the value of from
by just multiplying a ( x + y) with
.
How can you use polynomial identities to factor polynomials and simplify numerical expressions ?
![m to the power of 8 minus 9 n to the power of 10](data:image/png;base64,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)
How can you use polynomial identities to factor polynomials and simplify numerical expressions ?
![m to the power of 8 minus 9 n to the power of 10](data:image/png;base64,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)
Use polynomial identities to multiply the expressions ?
![left parenthesis 7 plus 9 right parenthesis squared](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADcAAAARCAYAAAB5LjRtAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAPUJuPDwAAAaJJREFUeNpjYBj64D8U/wLio0CsgkthFhB3DFJPdEDdhwswQeXPYZPUA+LjgzyWjkLdiQ/8wKXRAEt048LUBvZAfBjquG9AvAaIldDUGEPV4DPjILqgAdRzxIBQIJ5NZY+5AfEVIDaFJi8WIE4G4htALI6m9jDUk+hAEuoH9ABh6ALiHCIcIQ41nIOEzE4MOIPNUdCA7EITy8NSLqgB8RaoBzHADiC2JcIRm9CSLrU89wdPIYFeQFgC8V60GNsGxHy4DP8ExGwEHJABxLVkFNPEgLs4kpo8NP8hAzaoe2EA5DENckIO2ZKT0LxAC8+Bkt99aIHABMU+0Fj7hUX9LwIFH0meA5VAjiRUqOSUsiCP7YeWliC8ClohfyPgOYIAX7IMhUY9ua0HSgAHNMXgS5YEwW4gtsaRoW+QWIhQ03Og1NJMoEAhCHBVBQHYKkU6eq4ViKXRxHKwVA94Aa5KfDkQJ9LBc6CYCELKGtLQktkPi1pclThecBxLu+0lEAvSoc3oAi1M/kDz03IcbUhCzS+cYLg0nPFW1IO1ywPKZ2mkaAAAiPpuZM/W44UAAACIdEVYdE1hdGhNTAA8bWF0aCB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMTk5OC9NYXRoL01hdGhNTCI+PG1vPig8L21vPjxtbj43PC9tbj48bW8+KzwvbW8+PG1uPjk8L21uPjxtc3VwPjxtbz4pPC9tbz48bW4+MjwvbW4+PC9tc3VwPjwvbWF0aD6YBMINAAAAAElFTkSuQmCC)
Use polynomial identities to multiply the expressions ?
![left parenthesis 7 plus 9 right parenthesis squared](data:image/png;base64,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)
How can you use polynomial identities to factor polynomials and simplify numerical expressions ? 123 + 23
This can also be done by finding the cube of each values and adding them. But that might be time consuming. Hence, we use these identities.
How can you use polynomial identities to factor polynomials and simplify numerical expressions ? 123 + 23
This can also be done by finding the cube of each values and adding them. But that might be time consuming. Hence, we use these identities.
How can you use polynomial identities to factor polynomials and simplify numerical expressions ?
![open parentheses 11 cubed close parentheses plus 5 cubed](data:image/png;base64,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)
How can you use polynomial identities to factor polynomials and simplify numerical expressions ?
![open parentheses 11 cubed close parentheses plus 5 cubed](data:image/png;base64,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)
How can you use polynomial identities to factor polynomials and simplify numerical expressions ? ![27 x to the power of 9 minus 343 y to the power of 6](data:image/png;base64,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)
We use identities to speed up the process of multiplication and simplification. There are some basic polynomial identities that you need to by heart.
How can you use polynomial identities to factor polynomials and simplify numerical expressions ? ![27 x to the power of 9 minus 343 y to the power of 6](data:image/png;base64,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)
We use identities to speed up the process of multiplication and simplification. There are some basic polynomial identities that you need to by heart.