Question
How can you use polynomial identities to factor polynomials and simplify numerical expressions ? ![open parentheses 9 m to the power of 4 close parentheses minus open parentheses 25 n to the power of 6 close parentheses](data:image/png;base64,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)
Hint:
, where a and b can be real numbers, variables or multiples of both.
We are asked to factorize the given expression using a polynomial identity.
The correct answer is: (3m2 - 5n3)(3m2 + 5n3)
Step 1 of 2:
Here, the given expression is
.
It can be written as:
, where
.
Step 2 of 2:
Apply the polynomial identity
to factorize the expression
.
![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 9 m to the power of 4 close parentheses minus open parentheses 25 n to the power of 6 close parentheses equals open parentheses 3 m squared close parentheses squared minus open parentheses 5 n cubed close parentheses squared end cell row cell equals open parentheses 3 m squared minus 5 n cubed close parentheses open parentheses 3 m squared plus 5 n cubed close parentheses end cell end table](data:image/png;base64,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)
Hence, the answer is: ![open parentheses 3 m squared minus 5 n cubed close parentheses open parentheses 3 m squared plus 5 n cubed close parentheses](data:image/png;base64,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)
Hence, the answer is:
Factorization of polynomial of higher degree can be done by polynomial identities. This is used to reduce time and space.
Related Questions to study
Use polynomial identities to multiply the expressions ?
![open parentheses 4 straight x squared plus 6 straight y squared close parentheses open parentheses 4 straight x squared minus 6 straight y squared close parentheses](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAKIAAAATCAYAAAATZNbLAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAQ3ZOC+gAAAtpJREFUeNrtmj1IHUEQx4cg8oogPMQipAgEERGRB0FERNKIRbAPQUREEJFHivQpQpqHRUgbQgghnUWKICKIhTxEBLFIEUJArMReREQEM3v5Hyzn7n3tvrtdvYE/6L37mPvd7Ozs3BEVZzfQFWuX1U/umqu+Vgwt2gPWCuuQ3DdXfb3zDMUBrYIcvCR/TPa1BU4Vww4xHGHtFeTUc9aOJwBVvu6CV8XQPsPgh0bMyWYw55vaI1zraQdvfJS1xjrDSPwJGLZ8fcZqVwztM2xgZ509ZP22AHGAtQ7nOmVLgDaA/3tYs5rAMfG1DZgVQ3sMaZXVjDnhJ9aiIUThzAZuKs8KLI0NsQ4sZZwkX19HakHXGZIHDGmTNanZeYK1rQmIGmqix5HtfbiZmrRNODVo0ApIY+Jhv4r5vc461qzsTli9GXwdl7j4wJA8YBjUAd2KHcW2X6wnMQExjIvKJtLxtKavJMt2IP5JMWVtof6RbQpTURZfu8HNF4bkAUO61lywFZludDfelJbj86zvluGkBX6JuuQH64L+N1MPUN+Etqhor3xLyAI6u/KIIXnAUAlxRFF8xwXEGlZVR1J6Ngm8JOmOE43TF6wuTBdiWvwrPeQ6fJRH5bkmm5kGYsUwYyCqppV9uv1aJg7iJE661IFRmjYjnilqrXBFexKZWhpS9vmaw6c0U7NLDMkDhsFJJzKOqGih2kb6Xi8xEDfgS9LIE1PLe+nep3P4FC20XWdIHjBMbD0kBcQ71gL+nmN9LCkQxdTxUrF9ENlJXvmFRflpTp+a4OYLQ/KAYWIzNg5i2IGX7QNgFh2IItXvYLR2YdsYVq1TipXfF4MHnrWhXTZD8oBhYHukefcXA7EGZ+qaFD9eAkjRf/uM4vkaUFX9vbC5PJrjGrpXfBVDc4aFvrB3wRpYCeYxFz56uKsMA1um4j5hKtvesN7mOG41YVVbMTRneK9si9z+wvleMfwHYhBMBgi30WoAAAG4dEVYdE1hdGhNTAA8bWF0aCB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMTk5OC9NYXRoL01hdGhNTCI+PG1mZW5jZWQgc2VwYXJhdG9ycz0ifCI+PG1yb3c+PG1uPjQ8L21uPjxtc3VwPjxtaSBtYXRodmFyaWFudD0ibm9ybWFsIj54PC9taT48bW4+MjwvbW4+PC9tc3VwPjxtbz4rPC9tbz48bW4+NjwvbW4+PG1zdXA+PG1pIG1hdGh2YXJpYW50PSJub3JtYWwiPnk8L21pPjxtbj4yPC9tbj48L21zdXA+PC9tcm93PjwvbWZlbmNlZD48bWZlbmNlZCBzZXBhcmF0b3JzPSJ8Ij48bXJvdz48bW4+NDwvbW4+PG1zdXA+PG1pIG1hdGh2YXJpYW50PSJub3JtYWwiPng8L21pPjxtbj4yPC9tbj48L21zdXA+PG1vPiYjeDIyMTI7PC9tbz48bW4+NjwvbW4+PG1zdXA+PG1pIG1hdGh2YXJpYW50PSJub3JtYWwiPnk8L21pPjxtbj4yPC9tbj48L21zdXA+PC9tcm93PjwvbWZlbmNlZD48L21hdGg+m6VNugAAAABJRU5ErkJggg==)
Use polynomial identities to multiply the expressions ?
![open parentheses 4 straight x squared plus 6 straight y squared close parentheses open parentheses 4 straight x squared minus 6 straight y squared close parentheses](data:image/png;base64,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)
Use polynomial identities to multiply the expressions; ![left parenthesis 12 plus 15 right parenthesis squared](data:image/png;base64,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)
Polynomial identities are used to reduce the time and space while you solve higher degree polynomial expressions.
Use polynomial identities to multiply the expressions; ![left parenthesis 12 plus 15 right parenthesis squared](data:image/png;base64,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)
Polynomial identities are used to reduce the time and space while you solve higher degree polynomial expressions.
How can you use polynomial identities to multiply expressions ?
![41 cross times 39](data:image/png;base64,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)
How can you use polynomial identities to multiply expressions ?
![41 cross times 39](data:image/png;base64,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)
Use polynomial identities to multiply the expressions; ![open parentheses 3 x squared plus 5 y cubed close parentheses open parentheses 3 x squared minus 5 y cubed close parentheses](data:image/png;base64,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)
Polynomial identities are used to reduce the time and space while you solve higher degree polynomial expressions.
Use polynomial identities to multiply the expressions; ![open parentheses 3 x squared plus 5 y cubed close parentheses open parentheses 3 x squared minus 5 y cubed close parentheses](data:image/png;base64,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)
Polynomial identities are used to reduce the time and space while you solve higher degree polynomial expressions.
Use polynomial identities to multiply the expressions ?
![left parenthesis 2 x minus 5 right parenthesis left parenthesis 2 x plus 5 right parenthesis](data:image/png;base64,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)
Use polynomial identities to multiply the expressions ?
![left parenthesis 2 x minus 5 right parenthesis left parenthesis 2 x plus 5 right parenthesis](data:image/png;base64,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)
Use polynomial identities to multiply the expressions ?
![left parenthesis 3 straight x minus 7 right parenthesis squared](data:image/png;base64,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)
Use polynomial identities to multiply the expressions ?
![left parenthesis 3 straight x minus 7 right parenthesis squared](data:image/png;base64,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)
How can you use polynomial identities to multiply expressions ? 41
39
Polynomial identities are used to reduce the time and space while you solve higher degree polynomial expressions.
How can you use polynomial identities to multiply expressions ? 41
39
Polynomial identities are used to reduce the time and space while you solve higher degree polynomial expressions.
How can you use polynomial identities to multiply expressions ?
![open parentheses 2 x squared plus y cubed close parentheses squared](data:image/png;base64,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)
Any expression of the form can be expanded using the identity
rather than using the conventional and time taking method of multiplying them,
.
How can you use polynomial identities to multiply expressions ?
![open parentheses 2 x squared plus y cubed close parentheses squared](data:image/png;base64,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)
Any expression of the form can be expanded using the identity
rather than using the conventional and time taking method of multiplying them,
.
Prove the difference of cubes identity : ![a cubed minus b cubed equals left parenthesis a minus b right parenthesis open parentheses a squared plus a b plus b squared close parentheses](data:image/png;base64,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)
To prove an equation to be equal, we have to prove either the RHS is equal to the LHS or vice versa. The side should be selected based on the simplification process.
Prove the difference of cubes identity : ![a cubed minus b cubed equals left parenthesis a minus b right parenthesis open parentheses a squared plus a b plus b squared close parentheses](data:image/png;base64,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)
To prove an equation to be equal, we have to prove either the RHS is equal to the LHS or vice versa. The side should be selected based on the simplification process.
Prove the sum of cubes identity : ![a cubed plus b cubed equals left parenthesis a plus b right parenthesis open parentheses a squared minus a b plus b squared close parentheses](data:image/png;base64,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)
To prove an equation to be equal, we have to prove either the RHS is equal to the LHS or vice versa. The side should be selected based on the simplification process.
Prove the sum of cubes identity : ![a cubed plus b cubed equals left parenthesis a plus b right parenthesis open parentheses a squared minus a b plus b squared close parentheses](data:image/png;base64,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)
To prove an equation to be equal, we have to prove either the RHS is equal to the LHS or vice versa. The side should be selected based on the simplification process.