Question
Explain how to use a polynomial identity to factor
.
Hint:
, where a and b can be real values, variables or multiples of both.
We are asked to explain how to use a polynomial identity to factor ![8 x to the power of 6 minus 27 y cubed](data:image/png;base64,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)
The correct answer is: polynomial
Step 1 of 2:
Factoring a polynomial is the process of decomposing a polynomial into a product of two or more polynomials. This can be done with the help of identities, which would speed up the process and make it simple.
The given polynomial is
. It can be written as:
here
.
Step 2 of 2:
Substitute the values of
in the identity;
![table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell open parentheses 2 x squared close parentheses cubed minus left parenthesis 3 y right parenthesis cubed equals open parentheses 2 x squared minus 3 y close parentheses open parentheses open parentheses 2 x squared close parentheses squared plus open parentheses 2 x squared close parentheses left parenthesis 3 y right parenthesis plus left parenthesis 3 y right parenthesis squared close end cell row cell equals open parentheses 2 x squared minus 3 y close parentheses open parentheses 4 x to the power of 4 plus 6 x squared y plus 9 y squared close parentheses end cell end table](data:image/png;base64,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)
Thus, the factor is: ![8 x to the power of 6 minus 27 y cubed equals open parentheses 2 x squared minus 3 y close parentheses open parentheses 4 x to the power of 4 plus 6 x squared y plus 9 y squared close parentheses](data:image/png;base64,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)
Thus, the factor is:
We use identities to speed up the process of multiplication and simplification. There are some basic polynomial identities that you need to by heart.
Related Questions to study
Use polynomial identities to factor the polynomials or simplify the expressions :
![10 cubed plus 5 cubed](data:image/png;base64,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)
Use polynomial identities to factor the polynomials or simplify the expressions :
![10 cubed plus 5 cubed](data:image/png;base64,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)
Explain why the middle term
is 10x.
Explain why the middle term
is 10x.
Use polynomial identities to factor the polynomials or simplify the expressions :
![9 cubed plus 6 cubed](data:image/png;base64,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)
Use polynomial identities to factor the polynomials or simplify the expressions :
![9 cubed plus 6 cubed](data:image/png;base64,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)
How can you use polynomial identities to rewrite expressions efficiently ?
How can you use polynomial identities to rewrite expressions efficiently ?
Use binomial theorem to expand![left parenthesis 2 c plus d right parenthesis to the power of 6](data:image/png;base64,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)
Use binomial theorem to expand![left parenthesis 2 c plus d right parenthesis to the power of 6](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEEAAAARCAYAAACCecGyAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAPUJuPDwAAAiZJREFUeNrll0FEREEYx5+VlSQ67SGJdEiSZSUrHbok6bCiQzolkiQdolM6RTqubntIOkSSDh0ia2WtFcnqkCwdVvaQJdnDWmvZ/sN/GdPMe2/b3dT258d733vzZuab7/tmnmX9L42BW1AEFfJFa2D/j090n/NQNQQewLBd4xGQbJEVT3A+so7BhJuGful+HJyDPCiBFFj8xRMvAA+vAyCuPM9x/BmQBvPqB/x0giyROwugUwqnBG2/TQPgUbHF6YyqyiACuoCX10tygwOw7qKzPk1n9arSgG+EwKli21DqW1qKFIuLm5EbXLvJF6posHfSmVmmzwXoboITRD9h8MGxiJQ9BNvKe0EQle4jynhENNzLDfI0OimoSZvqwETl3ZPCbQvMNtgJYiVjDGMP+9rlN0LKu17OSy78x2zTRsdNW0q+OKkd3LFg6ral8A+kw45mxauh3qOxl5R7URhfwJuuMDo5QYTRJZgyrE7OZehb0iHFDpNepUIt918wvF+qZTXs0qGfDhgwPB/VbEfNiAQ/dyzdKTCmsavp4KgbQ5gPsqB02LSdYRFsthPmwInGLrbsI0P9itYyEN0W6QNnLCJ2Envx8w84IcTxqBKOWdHY1zkv19Idlq4YCW6U4s7Qxuq7aYisepwgovFJOtUGOMYso9FyOCy5UlI5b9dSuHqZr2UeppabdDL0MfWKHO8keOfOpUbnt+pUq/9AudZqC/xKHxhqhFGfHiODGpLHeFEAAACSdEVYdE1hdGhNTAA8bWF0aCB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMTk5OC9NYXRoL01hdGhNTCI+PG1vPig8L21vPjxtbj4yPC9tbj48bWk+YzwvbWk+PG1vPis8L21vPjxtaT5kPC9taT48bXN1cD48bW8+KTwvbW8+PG1uPjY8L21uPjwvbXN1cD48L21hdGg+r/68FwAAAABJRU5ErkJggg==)
Use polynomial identities to factor the polynomials or simplify the expressions :
![open parentheses 1 over 16 x to the power of 6 minus 25 y to the power of 4 close parentheses](data:image/png;base64,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)
Use polynomial identities to factor the polynomials or simplify the expressions :
![open parentheses 1 over 16 x to the power of 6 minus 25 y to the power of 4 close parentheses](data:image/png;base64,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)
Use binomial theorem to expand .![left parenthesis x minus 1 right parenthesis to the power of 7](data:image/png;base64,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)
Use binomial theorem to expand .![left parenthesis x minus 1 right parenthesis to the power of 7](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADgAAAARCAYAAACIJW/gAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAPUJuPDwAAAUZJREFUeNpjYBge4D8OjAGygLhjEHukA+pGQiAWiBejC+oB8fEhEFtHoW7FBbiA+AIQC2PTaDAIPKAGxLVQR2IDxkB8GI/+VmgMogADqAcHAwAlrTRceQgKDkM9ig6UcPmjC4hzBmHBgQvk4Sgr1gGxLTYNO3BIJOMwqBkqN1AetATivViSLs5U+AmI2XDInQNicSR+IhBPGeAYZIO6GRnMh7oNK/iDxzAPIO6Dsl2whNxAeBAEfqHxn2IrOYnxIAjsBmJ7XMUvkZUu3gqYCh7EC/AlURAogBqoN0gKGWxJlGAMWeOQA4mvgSbTyEHiQUtSswquagJUr2yCtg54gPgMEUmUHh7MgbqZaICtohcF4m1oHvLB1sYbAA/iqujxguNIeYwNWmlKY1G3HFqy0rNnQEpTDScYLo1tvCBjkHeXuqDtVKIBADjUVsSNQZ7lAAAAj3RFWHRNYXRoTUwAPG1hdGggeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzE5OTgvTWF0aC9NYXRoTUwiPjxtbz4oPC9tbz48bWk+eDwvbWk+PG1vPiYjeDIyMTI7PC9tbz48bW4+MTwvbW4+PG1zdXA+PG1vPik8L21vPjxtbj43PC9tbj48L21zdXA+PC9tYXRoPoRq1bIAAAAASUVORK5CYII=)
Use polynomial identities to factor the polynomials or simplify the expressions :
![64 x cubed minus 125 y to the power of 6](data:image/png;base64,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)
Use polynomial identities to factor the polynomials or simplify the expressions :
![64 x cubed minus 125 y to the power of 6](data:image/png;base64,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)
How are Pascal’s triangle and binomial expansion such as (a + b)5 related?
You can find the expansion of (x + y)n using both Pascal’s triangle and binomial expansion.
How are Pascal’s triangle and binomial expansion such as (a + b)5 related?
You can find the expansion of (x + y)n using both Pascal’s triangle and binomial expansion.
Use binomial theorem to expand .![open parentheses s squared plus 3 close parentheses to the power of 5](data:image/png;base64,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)
Use binomial theorem to expand .![open parentheses s squared plus 3 close parentheses to the power of 5](data:image/png;base64,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)
Use polynomial identities to factor the polynomials or simplify the expressions :
![216 plus 27 y to the power of 12](data:image/png;base64,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)
Use polynomial identities to factor the polynomials or simplify the expressions :
![216 plus 27 y to the power of 12](data:image/png;base64,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)
Explain why the middle term
is 10x.
In the Binomial Expansion's middle term, in the expansion of (a + b)n, there are (n + 1) terms. Therefore, we can write the middle term or terms of (a + b)n based on the value of n. It follows that there will only be one middle term if n is even and two middle terms if n is odd.
The binomial expansions of (x + y)n are used to find specific terms, such as the term independent of x or y.
Practice Questions
1. Find the expansion of (9x - 2y)12's coefficient of x5y7.
2. In the expansion of (2x - y)11, locate the 8th term.
Explain why the middle term
is 10x.
In the Binomial Expansion's middle term, in the expansion of (a + b)n, there are (n + 1) terms. Therefore, we can write the middle term or terms of (a + b)n based on the value of n. It follows that there will only be one middle term if n is even and two middle terms if n is odd.
The binomial expansions of (x + y)n are used to find specific terms, such as the term independent of x or y.
Practice Questions
1. Find the expansion of (9x - 2y)12's coefficient of x5y7.
2. In the expansion of (2x - y)11, locate the 8th term.
Use binomial theorem to expand
.
Use binomial theorem to expand
.
Use polynomial identities to factor the polynomials or simplify the expressions :
![4 x squared minus y to the power of 6](data:image/png;base64,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)
Use polynomial identities to factor the polynomials or simplify the expressions :
![4 x squared minus y to the power of 6](data:image/png;base64,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)
How can you use polynomial identities to rewrite expressions efficiently ?
Polynomial identities are equations that are true for all possible values of the variable and numbers.
How can you use polynomial identities to rewrite expressions efficiently ?
Polynomial identities are equations that are true for all possible values of the variable and numbers.