Question
Use polynomial identities to multiply the expressions;
![left parenthesis 12 plus 15 right parenthesis squared](data:image/png;base64,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)
The correct answer is: Hence, the answer is: 729.
ANSWER:
Hint:
Let a and b be any real number, variables or multiples of the both, then we have: .
![left parenthesis a plus b right parenthesis squared equals a squared plus 2 a b plus b squared](data:image/png;base64,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)
We are asked to use the polynomial identities to multiply the given expression.
Step 1 of 2:
The given expression is
.
Here,
.
Use the identity
for the expression ![left parenthesis 12 plus 15 right parenthesis squared](data:image/png;base64,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)
Step 2 of 2:
Now, substitute the value in the equation
to get the value of
:
.
![equals 144 plus 360 plus 225](data:image/png;base64,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)
=729
Hence, the answer is:729 .
Note:
Polynomial identities are used to reduce the time and space while you solve higher degree polynomial expressions.
Related Questions to study
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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)
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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)
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.
Use polynomial identities to multiply the expressions ?
![open parentheses 6 minus y cubed close parentheses squared](data:image/png;base64,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)
Use polynomial identities to multiply the expressions ?
![open parentheses 6 minus y cubed close parentheses squared](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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)
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,iVBORw0KGgoAAAANSUhEUgAAAKYAAAATCAYAAAAaj3axAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAQ3ZOC+gAAAupJREFUeNrtms9nnEEYxx+rVtUqsaqqhxJROcQK1UNVVamKiBUhooeKKBEVtXqPHqpED9FDrz1URanooeJVqqIiolRE5BClKv9ADjnEirJ95vV9eY2Z3bfvr52nnS9f9v2x8858PDPzzLwvUbnqwKfsLfYQua14fXfZ9xytm2eZkyrsR+wdkqNh9qFn2R+WqoHLJVawLQimCoCj2PEyeHmWBbNssLdLrNxt9lchIOvsV+w57fwWuHmWxbEML4zGju+wA/YJeuMP9ksUnFWX8LzBAnMZk7OU98Bw7Rp707MsjuUoGhfXBnuKXdXu+5SxsVfZ6wBaVJKdt86zn7PnDdc2AdWzzJ8lvWAvJiz4OGPvDlC5oiB1qDiZ8rjHWi4pgSUJYRn23FsJClPTxYHh/ENLov8M1yIFWJGRA4H5mj1jON9Cj9al+Hw3nL/B/iKMJQlhGfbcapcHX0SyqnKjccs9O7gv0hyS3F45S78CcxajW1zn0Ma6Vt82Au6KoZyqNvJJYElCWNLvhMlvq0vlxtgr+H1Xj/wSIUWbuMcYVZ6wa4b7VFC81849ZS+lqNupMJYkhKUVZqQB9iT7G7YmbPqM67s5rDg7CdxNZ5BIL2HKHDZsW+zHji+wf7LPFhSYnmWKwOw1/USqAahNLRTccCwRV6+/TPt8J7HfKz1GMZv+dip3gSUJYRn2zpsZVlOE/6+hUvcdXCG2LdsTg7Dq4ZUU5eoJuwSWJIRl4i2OBhJa0wrzIxLeGlZcdYcCcwSwdL1hN9mrZN70TaJFLfGXwJKEsDRuCqvhehr5hYp+9fbiF1ZgpOUTgQZvgv22T4H5AT2vAo8B5JRlulRbHXsZ6pVkg901liSEZahtLZ9Re03rGLaVNwBJzwlU5S8bHvIODSlb0xiJ1CLkCKvF65Z7JxHwzZTPsr2S9CzzY1n6hwcuqJmxza58xPEvswy1QOV+qtVvBZiq0kjlQvNdrnuW+bH8rzSC6dXLQZZ/ANlPRYpc5j4fAAABZHRFWHRNYXRoTUwAPG1hdGggeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzE5OTgvTWF0aC9NYXRoTUwiPjxtZmVuY2VkIHNlcGFyYXRvcnM9InwiPjxtcm93Pjxtbj4zPC9tbj48bXN1cD48bWk+eDwvbWk+PG1uPjI8L21uPjwvbXN1cD48bW8+KzwvbW8+PG1uPjU8L21uPjxtc3VwPjxtaT55PC9taT48bW4+MzwvbW4+PC9tc3VwPjwvbXJvdz48L21mZW5jZWQ+PG1mZW5jZWQgc2VwYXJhdG9ycz0ifCI+PG1yb3c+PG1uPjM8L21uPjxtc3VwPjxtaT54PC9taT48bW4+MjwvbW4+PC9tc3VwPjxtbz4mI3gyMjEyOzwvbW8+PG1uPjU8L21uPjxtc3VwPjxtaT55PC9taT48bW4+MzwvbW4+PC9tc3VwPjwvbXJvdz48L21mZW5jZWQ+PC9tYXRoPmL2uGcAAAAASUVORK5CYII=)
How can you use polynomial identities to factor polynomials and simplify numerical expressions ? ![x cubed minus 216](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 ? ![x cubed minus 216](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.
Use polynomial identities to multiply the expressions ?
![open parentheses 8 minus x squared close parentheses open parentheses 8 plus x squared close parentheses](data:image/png;base64,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)
Use polynomial identities to multiply the expressions ?
![open parentheses 8 minus x squared close parentheses open parentheses 8 plus x squared close parentheses](data:image/png;base64,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)
Use polynomial identities to multiply the expressions ?
![open parentheses x squared plus y to the power of 6 close parentheses squared](data:image/png;base64,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)
Use polynomial identities to multiply the expressions ?
![open parentheses x squared plus y to the power of 6 close parentheses squared](data:image/png;base64,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)
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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)
Factorization of polynomial of higher degree can be done by polynomial identities. This is used to reduce time and space.
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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)
Factorization of polynomial of higher degree can be done by polynomial identities. This is used to reduce time and space.
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 ?
![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,iVBORw0KGgoAAAANSUhEUgAAAHQAAAAQCAYAAADH/1trAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAOJ5y/mQAAAd5JREFUeNrtmTFLw0AYhg+RIuIiIg4OgjiISCmIg4i4SOngJDgUJxFEpIi4O7kUh05ujg6CiIiICOIgpRRBRJxE8C84OJUinG/CK5zlkh42Xq+JHzzQ5HJJn97lvi+pED9jExRF50eRLkGRCM80qIr4RIVOSfX0GzLK9iw4BR+gDp7AigMCMgQ1pkA5qZ4ZiqpxB/Kgj9sTPCbvgKhplCmcNE+xDwoGHUfAcweJbjXkShc8pQVPcQ3mDDvXNPvWAoqMPba1a0BnwK1jntKCp58/UoYdKwFtj2BI2V4FB22+Q1N0c8lTWvAUnwadesA9iwhd5ECJnxcaZ0zEA1qnwBXYUfKfLuqOeUoLnk1F+8E5yDY57gbMs1IcaKGKkwbi3SwEdsELGI9gQOPiGboUjVJyzGBWbfPEacuFUpbVaitLri1PacHTn3G6JcabDYeg1+Bi389zpTaV/DWDYsEFT2nBU1vOe4n/hLd8s/Bm9wV/EG+dfzBYiqKMSfCm2V+gm0ue0oKn9oH7MmS9VmOQSVsVWwRHfzR4Z5yRXSRHyaVfvliw7SktePpRbcgJJkk8xQsPay5yzC8RdSyDVxY477y7pjXHBb36S4rn/8v5mHn6sSHi8beSl0/WQ9pj6/kFI5XXaePiS1YAAAC+dEVYdE1hdGhNTAA8bWF0aCB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMTk5OC9NYXRoL01hdGhNTCI+PG1vPig8L21vPjxtbj4yPC9tbj48bWk+eDwvbWk+PG1vPiYjeDIyMTI7PC9tbz48bW4+NTwvbW4+PG1vPik8L21vPjxtbz4oPC9tbz48bW4+MjwvbW4+PG1pPng8L21pPjxtbz4rPC9tbz48bW4+NTwvbW4+PG1vPik8L21vPjwvbWF0aD63fIrFAAAAAElFTkSuQmCC)
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,
.