Question
Explain why the middle term
is 10x.
The correct answer is: Here, a=x and b=5.
ANSWER:
Hint:
The binomial expansion is
, here
.
We are asked to explain why the middle term
is 10x.
Step 1 of 1:
The given expression is:![left parenthesis x plus 5 right parenthesis squared](data:image/png;base64,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)
Here, a=x and b=5.
Here, the middle term is 10x because while expanding you have the form,
![left parenthesis x plus 5 right parenthesis squared equals 2 C subscript 0 x squared plus 2 C subscript 1 left parenthesis 5 right parenthesis left parenthesis x right parenthesis plus 2 C subscript 2 5 squared](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAARMAAAAVCAYAAABlhwoRAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAQ3ZOC+gAABFNJREFUeNrtXMFnHUEYH09EVJWKqKgqT0RVRageIiKXqIieSg/RU5Wqiqgq1UNPuURUT71VVeVQqioqqlRVRDylKnqoKD30H4iKqljh9fu878lmzb6Z2flm5u2++fHjvey+3dnf95tvZ77diRARVUKTmAC3gCNRkojosXiBNqgB7wC/Re9HhPIYblzu4gtYpjbGTqSH/Qpek6lHVZ4p4nnVMXveY2PARgkav0Vt7bVOZIpp4EbFrqmoR/M8Y+N5HR/2rMdQnPEumKrImMZF4GaATjQJfAPco+nUNvC6xu/qwBXa/x/wN3AJeNKhlsMUz3rFdMh61NYzNp5X+bBs+upqqfTYOG0IXffQxSYF02cnwgQ1DzxO38/TueY7/OY+cB04RVMwxBngXeALRzqO0jmHHR0/lA4yj9p4hsPzeT4so75NLo9hRlsInCRMLmZRMs913YlkOAv8nrPtAfCZ54SM1/4eeMJxMg+hg8yjNp7h8PyiQb2l2/VtcnnsA2W1LG7miLVE20IlkwngJ6ZOZIv9nMz9C9jHdA7dOKAG5zyMDEPoIPOojWc4PJ89Zpn11W2f0mM4P+vP2YZPRk6lvt8APg08MumnNhfpRE0NmhhUNlR+TENMTujEweZabMzuQweZR208w+H57DHLrG/Tov8cwUGHH88Cn9DnGYNMXKSRCQUHk8O91LxRhoSpExXFAPCLaBXMsvgh+N91sY2DK7P70uGA2TNcnk8qoq+plkaBSuOjaD0lwUrxoOM7fx8VtR4BdzqMOBIRDlghXwNelmyrCXePpk3i4GOE5lOHA2bPcHk+qYi+ploWmuYIGkqhaGMFTVsUKOKG5fCSO8h1CvCI5nCaEzZx4I6Tbx1UHjX1DIfnXU5zQvpMpaUyC0/mbGs/98Zh37znZJJXeDIpfHECszRWzo8p9vursU8aV+gusEOfXcSBM06udBilO+K2oUeLeIbD864KsC70LbrkxHj0k/eYDLPjO2owzp++Fhhe2ySTC6JVrc5igdrsE1iQey30KufPRevZvw4mKPsPEj/T37jjwBUnVzogVoG3ctqi+yhX1zMcnjfxYTfo254e6S45ydOyI2Qv8AxRIWYwcwdddWTSt9SJasRZupCrkn1dvCykwrrB/BHfC9gFPhSHbyAOkH5ogJnUvms0N29jiszMHQeuOLnSQdUWmUdtPMPheRMfdpO+shGHiZZKNFLzw346+GnJfq/oRNy4BvxJhbFdys6XJPu5eo3ZptYiAw4jX9K8Fvf5Q0PnOUktoJa5c+wFjEMoHXQ6XiNTw7D1jI3nXfnQh76yJSe6WmqhSgv9ygSZSRLR22gyezQu9DuEyyUnR3BbdPe/IFihOXWVkB2Z9An3VfqyJpMiHlV5pojny+rDEEtOIjwiWzOZpoDHZBLBPSIJteQkwhOw6IVPcLCANkRz2bmYTCKYYbNuK6JEwOo7Fr+w8LXY40nE97KIqC0T/gO6igLpgsmpbwAAAaV0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+KDwvbW8+PG1pPng8L21pPjxtbz4rPC9tbz48bW4+NTwvbW4+PG1zdXA+PG1vPik8L21vPjxtbj4yPC9tbj48L21zdXA+PG1vPj08L21vPjxtbj4yPC9tbj48bXN1Yj48bWk+QzwvbWk+PG1uPjA8L21uPjwvbXN1Yj48bXN1cD48bWk+eDwvbWk+PG1uPjI8L21uPjwvbXN1cD48bW8+KzwvbW8+PG1uPjI8L21uPjxtc3ViPjxtaT5DPC9taT48bW4+MTwvbW4+PC9tc3ViPjxtbz4oPC9tbz48bW4+NTwvbW4+PG1vPik8L21vPjxtbz4oPC9tbz48bWk+eDwvbWk+PG1vPik8L21vPjxtbz4rPC9tbz48bW4+MjwvbW4+PG1zdWI+PG1pPkM8L21pPjxtbj4yPC9tbj48L21zdWI+PG1zdXA+PG1uPjU8L21uPjxtbj4yPC9tbj48L21zdXA+PC9tYXRoPq8q6okAAAAASUVORK5CYII=)
![equals x squared plus 2 left parenthesis 5 right parenthesis left parenthesis x right parenthesis plus 5 squared](data:image/png;base64,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)
![equals x squared plus 10 x plus 25](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHkAAAARCAYAAAD5XQN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAQ3ZOC+gAAAiBJREFUeNpjYBgY8B+KfwHxUSBWYRgFwxYwAXEWEJ8bDYrhD36MBsHwBvZAfHA0GIYvkITWyUo0tEMNiGuB+AIeNXxAPA2IT0LxTKjYQAJrIF4DxJ+gbReQ+6PxtG+w4QEHoMDfAo1oWoLFQJxGwNN7gdgPie8DFRtIACrdIoGYB8rXgmaISCyRPGhz8DYycwu5nsKlLxSIJ2ERn4AlQKnVsyAXyAPxpYGO5GQg7sAi3gyVgwFQBGvQOZBw6QMViW5YxB2hcuT4j1aRjK2R+p/cwCCE8QFQd0gciZ8IxFOIsGOgIvkdELNhEQeJvSTTf7SKZEtokT3gxbUHEPdB2S40qNuoHcl/8Oj5RQP/ket+Dmij0BqLeb+gDTRQCVmEVI/TFOyGdo1ALUJhKkQqJSULJZH8gwr+o4b7BYF4A45qBQZYgNgY2pO4Qag6pIajCqCpS28QNVxw6XuJo7jmwFFcU+o/Ut2vBI1gUoZ+3Wg99gDr3/UNstYpvoaXB46AWkMD/5HiflBunA3EXFRooFENgFLdJqijQPXCGSoU17SO5CBoQKKD+VgikRr+I9b9oMbdKmgxTCrQAeK7tIhgUWjFL4w2qLB4kEcyCOyHFsEs0KK7GktxRy3/Eev+LUR2M9dBW91MUOwBjeAgakcwG9QyaSxyy3EUh/QCxLQpQA2budAi7hsDZFiTZ4D9R2ybCDSYcwvagHwHzf2mDKNgZAEARMy+41PlPJMAAACedEVYdE1hdGhNTAA8bWF0aCB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMTk5OC9NYXRoL01hdGhNTCI+PG1vPj08L21vPjxtc3VwPjxtaT54PC9taT48bW4+MjwvbW4+PC9tc3VwPjxtbz4rPC9tbz48bW4+MTA8L21uPjxtaT54PC9taT48bW8+KzwvbW8+PG1uPjI1PC9tbj48L21hdGg+rTVoFgAAAABJRU5ErkJggg==)
Note:
The expansion of
is one of the standard and widely used polynomial identity.
Related Questions to study
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,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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 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,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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,iVBORw0KGgoAAAANSUhEUgAAAGcAAAATCAYAAACN8hMuAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAQ3ZOC+gAAAttJREFUeNrtWE9kXEEYf9ZasSrEWlE9hBWxaq1QERXRS0RURYWoiqoIOUQP0Xv1UKV62FMvUTlURIhaVbVK9RARUVZEVUXoIYccIuQQFWuV9Pv4LZ/pzO7My3vdYffHj32z7837Zr5/v3lB0H5cgnXiPnEy6DyMEreINbEf3iFPPOowx9wk7hELvhuaIJ51mHPeEcd9NzJDfEOc7zDnnBLnUDEOibOmG0eIm8Rz1L+PxDtNJr4XUX1s1NlHMW/EEPEZepsOY8T3WH+jB841sVdHV/whviX2ElP4/U+ALsIZQ7juhWHbhkmvEX9G2Lz4fS9hR1xYw/wmm7kpP8TaGv1gB2Oqc6LCIcq53NcjtSlVHSddIS7EoCxq/0kh2mKA+D1G53Cm9InrlOqLFU10NAOn/9cWhrLjXmnGX+A/HcZDBEncztEFjO3zq8QHmvFlVAlGEaKAK0cSfXdK3nxAvG75whQiacDCUJaI/eJ6Hi/X1W/egM9iXl+ccxulLczzj4mvlbE0SllGjHH7+EU80QmCGnpNmXiBZlg1NEPOhieWhnIElPB7QmSbDwdfG/QQv6FSqM/XIRwqxKeiT0nchcCSeA5R4mTsHiZLokGNwcNL4r5iiCj6AsW3r0TLVVTdVdWSzX3cBz60+GLBe3ULm32AQ7R6PPghrrPIkB6XRXME3NCMDxOPxTVH0aDjQpcRZUWPzhatbM7BMYMOc05C7am4EL9L2A8nVBQ5J1F3iFzTuaHkKDja6Zw8FFQ6IqW5DWfnkDUJ10mXDKoij2wJs9Aczk1p1ONqBGUtbuf0o0ckQ8xZwObrPs9ME9fDHrJTSMkFYdgoVNlEiIVmkY0Z5WvCmufO+aTpGzqUoeIS4BQcM2Mo66uas5ITskjn3/iksBXYfZC71Di6bOhhG6qGb4NTmpVi25I9C7HE+3SGbBsxvPM+np8OuvAO7JTd7jb4iQpKYBeeoYA+Fgp/ARl+yL1zZP9rAAAAqXRFWHRNYXRoTUwAPG1hdGggeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzE5OTgvTWF0aC9NYXRoTUwiPjxtbj42NDwvbW4+PG1zdXA+PG1pPng8L21pPjxtbj4zPC9tbj48L21zdXA+PG1vPiYjeDIyMTI7PC9tbz48bW4+MTI1PC9tbj48bXN1cD48bWk+eTwvbWk+PG1uPjY8L21uPjwvbXN1cD48L21hdGg+M6qY8gAAAABJRU5ErkJggg==)
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,iVBORw0KGgoAAAANSUhEUgAAAFwAAAATCAYAAAAd4WrhAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAQ3ZOC+gAAAitJREFUeNrtV8FHREEcfp6VrESyVjrEWmulw16SJIl02EO6rKwkiZU97T+QTl06dE86JIkkydp7VhIrWR2ydOzQNVlrxes3fIfnmbdvZt7M6/I+PvbNjLfffPOb3+/3LCuGKnLEfeIrZ84B+8RHYja2KzwuiBUY6webWCW+xHbpgyOwphfbFJ3hS8QH7+AC8Yb4jbzD8tKmYv5yY5Z4jfeyU76HAJ0Q1e4E0IThE8jhGe8EO4EycQTP01hYDpG/KjA4h+dRGNHUbLisdi9KxFMDhrN912G6EKaIbcU/Y5tuRZAbw2hnSCMAhjXrZCY3EGRS6CmaciIRYSYMFy1U7AYWPGNnxA3O2hrxUFAnMzsvK3geV1PFlHeZq2TAcBHte6hDXmwTjzxjSWKHOB5QE0TmuGBX7BkFScWUHnLYLbGLYtYKKMS6DBfRPoU1Cc5cEYXejQOfw9GCMeIdcTWEKQ6a/SI2ZcOADj4ELIUuwtGonRXaZZ85FsVvrucU8SNknvdFBoKzIaOQtWiTnHGWLz8NRbio9hJy7CB0Xb+Pkb+1I4/2KKnh2jcQ1Tz0DRguqt1GfSkErGviADOIblu32WnkrYSmwlb1qfR55E6dhstoX+d99XFwTlwjXhK3TER3XaWNGWDKEDa26zJiDr3ximbDZbRfEXcE1tXQHrYtQ5AtVCLrUrjmP8RfHMDiP2v/QmEVuQkOojxGBGBGP8U2RIcGPp5iRIAZ1AUj+ANBkK+WPOsIzgAAAIJ0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW4+MjE2PC9tbj48bW8+KzwvbW8+PG1uPjI3PC9tbj48bXN1cD48bWk+eTwvbWk+PG1uPjEyPC9tbj48L21zdXA+PC9tYXRoPl4oa1YAAAAASUVORK5CYII=)
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,iVBORw0KGgoAAAANSUhEUgAAAD8AAAATCAYAAAAnMdWSAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAQ3ZOC+gAAAcJJREFUeNpjYKAd+A/Fv4D4KBCrMAwcMAfig0D8A8lddAFMQJwFxOcGyONaULt1BjDwwaE+EGAhENsOpMftocluIMBrII4G4odAfAuIQwlp8KFivpCE5nmlAfL8HyCeDcR8QMwGZSfiUswDxNeo5Hk1IN4CDYCBAreg5Q6y/x7iUjwTiJOp4HmQh7dBQ3wgASimBZH4oNg/g02hNRDvRaqqsAFQwHRgEW+GysEAyOMaNPbYXCAOxyJeAMStULYetNADRQILEE8BYg90DaAQuQTE8gQ8zwCtOsSR+IlQQ7HV8/9pWLfGA3EXmhgXNKkLI4mBCry7QPwSV4EHis0cNMfjAqCQ64OyXZBSC72BFxCvQhOrB+JaUgzRg5bIDER6HgR2Q6uwC2ihTElrEB/GBkD2XkHii0JjmIMUy09iaX4S8nwBtOmqN8AF2jckdh/UXVQNeWwF4xqoZZED7PnD0DaEEjTWmajVMcEGQJZsghYsPNBqQ3gAPQ8qyf2AeCkQx1KzV4YORKFVmDBaa3DxAHq+AFrlXaJ2lxS9KlwHxNJY1C7HVnfSCQRA3erHMAIByNPHGUYoAGVDy5HocR1ox4ksAAARiGnTBjvoPQAAAJx0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW4+NDwvbW4+PG1zdXA+PG1pPng8L21pPjxtbj4yPC9tbj48L21zdXA+PG1vPiYjeDIyMTI7PC9tbz48bXN1cD48bWk+eTwvbWk+PG1uPjY8L21uPjwvbXN1cD48L21hdGg+8/qD2AAAAABJRU5ErkJggg==)
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.
Use binomial theorem to expand (2c + d)6
The expansion of the expression can also be found using the Pascal’s triangle. But, it is necessary to remember the values of the triangle to write down the expansion.
Use binomial theorem to expand (2c + d)6
The expansion of the expression can also be found using the Pascal’s triangle. But, it is necessary to remember the values of the triangle to write down the expansion.