Maths-
General
Easy
Question
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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)
The correct answer is: Thus, the expansion is: .
Hint:
Let a and b be variables or real numbers, then:![left parenthesis a plus b right parenthesis squared equals a squared plus a b plus b squared](data:image/png;base64,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)
We are asked to explain how you can use the polynomial identities to multiply expressions.
Step 1 of 2:
Here, the given expression is
, where .![a equals 2 x squared straight & b equals y cubed](data:image/png;base64,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)
So, we can use the identity
to find the expansion of
.
Step 2 of 2:
Apply the polynomial identity,
to get the answer of
Thus, we have:![open parentheses 2 x squared plus y cubed close parentheses squared equals open parentheses 2 x squared close parentheses squared plus 2 cross times 2 x squared cross times y cubed plus open parentheses y cubed close parentheses squared](data:image/png;base64,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)
![equals 4 x to the power of 4 plus 4 x squared y cubed plus y to the power of 6](data:image/png;base64,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)
Thus, the expansion is: ![4 x to the power of 4 plus 4 x squared y cubed plus y to the power of 6](data:image/png;base64,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)
Note:
Any expression of the form
can be expanded using the identity
rather than using the conventional and time taking method of multiplying them,
.
Related Questions to study
Maths-
Use polynomial identities to multiply the expressions ?
![left parenthesis x plus 6 right parenthesis squared](data:image/png;base64,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)
Use polynomial identities to multiply the expressions ?
![left parenthesis x plus 6 right parenthesis squared](data:image/png;base64,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)
Maths-General
Maths-
Use polynomial identities to multiply the expressions ?
![left parenthesis x minus 9 right parenthesis left parenthesis x minus 9 right parenthesis](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGAAAAAQCAYAAADpunr5AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAOJ5y/mQAAAbBJREFUeNpjYEAFWUDcwTB4QQfUjbjAkHa/HhAfZxj84CjUrcPN/WAJgwF2nD0QHwbiH0D8DYjXALESmhpjqJph5X4DqAcGErgB8RUgNgViJiBmAeJkIL4BxOJoag9DPTJc3M/QBcQ5A+yBM1hSCwiEQt2HDPLQyvqh7n6GHUBsi0VzMo5KrRkqR03wB4c4KDWdQxOzBOK9w8j9DJ+AmA2HAefQslAiEE+hQQq6i54toUAeWp4iAzaom4eL+3HGHgh4AHEflO2CHnNUBKCseh9akTFBsQ80AH9hUf9rGLkfrwdAYDfU4AtALExA7X8iML5WxH5oKwKEVwGxCpYUREoEDAX3483CIFAA1aA3AJUbBxCfpKAIGgruB6cQaxwGWEPbs6BsHDkAHnCEVpr4KrGh7n6czThQs2oTEHMBMQ+0qSVMZw+0ArE0mlgOWtNuqLsfa0dGFIi3oTkYVKksppFDQSkiCKkoATm6Foj9sKglpiM2lNwPBseRykiQIeuwxBwILIe2LKgNXKAV2B9o+bgcR5mNayhiqLt/dDBuoAfjQCCDYXAP54LKzTQ88kPO/QAd96hD6wS9kAAAALF0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+KDwvbW8+PG1pPng8L21pPjxtbz4mI3gyMjEyOzwvbW8+PG1uPjk8L21uPjxtbz4pPC9tbz48bW8+KDwvbW8+PG1pPng8L21pPjxtbz4mI3gyMjEyOzwvbW8+PG1uPjk8L21uPjxtbz4pPC9tbz48L21hdGg+T1tDFgAAAABJRU5ErkJggg==)
Use polynomial identities to multiply the expressions ?
![left parenthesis x minus 9 right parenthesis left parenthesis x minus 9 right parenthesis](data:image/png;base64,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)
Maths-General
Maths-
Prove the polynomial identity ![x to the power of 4 minus y to the power of 4 equals left parenthesis x minus y right parenthesis left parenthesis x plus y right parenthesis open parentheses x squared plus y squared close parentheses](data:image/png;base64,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)
Prove the polynomial identity ![x to the power of 4 minus y to the power of 4 equals left parenthesis x minus y right parenthesis left parenthesis x plus y right parenthesis open parentheses x squared plus y squared close parentheses](data:image/png;base64,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)
Maths-General
Maths-
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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)
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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)
Maths-General
Maths-
How could you use polynomial identities to factor the expression
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How could you use polynomial identities to factor the expression
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Maths-General
Maths-
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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)
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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)
Maths-General
Maths-
After diluting salt water, the concentration of salt in the water is given by the function
, where x is the time in hours since the dilution.
b. After how many hours will the concentration of salt in the water be ?Round to the nearest hundredth.
After diluting salt water, the concentration of salt in the water is given by the function
, where x is the time in hours since the dilution.
b. After how many hours will the concentration of salt in the water be ?Round to the nearest hundredth.
Maths-General
Maths-
Which function has a graph with a vertical asymptote at x = 3? Select all that apply .
Which function has a graph with a vertical asymptote at x = 3? Select all that apply .
Maths-General
Maths-
Which function has a graph with a horizontal asymptote at y = -1.
Which function has a graph with a horizontal asymptote at y = -1.
Maths-General
Maths-
After diluting salt water, the concentration of salt in the water is given by the function
, where x is the time in hours since the dilution .
a. What is the concentration of salt in the water after 4 hours?
After diluting salt water, the concentration of salt in the water is given by the function
, where x is the time in hours since the dilution .
a. What is the concentration of salt in the water after 4 hours?
Maths-General
Maths-
A Software CD can be manufactured for
each. The development cost to produce the software is
. The first 200 CDs were used by testers to test the functionality of the software and were not sold.
a .Write a function f for the average cost , in dollars , of a sellable software CD , where x is the number of sellable software CDs
b. What are the vertical asymptotes of the graph ?
c. Graph the function
A Software CD can be manufactured for
each. The development cost to produce the software is
. The first 200 CDs were used by testers to test the functionality of the software and were not sold.
a .Write a function f for the average cost , in dollars , of a sellable software CD , where x is the number of sellable software CDs
b. What are the vertical asymptotes of the graph ?
c. Graph the function
Maths-General
Maths-
Amaya made 10 three point shots out of 25 attempts. If she then goes on to make consecutive three point shots, her success would be given by the function![f left parenthesis x right parenthesis equals fraction numerator x plus 10 over denominator x plus 25 end fraction](data:image/png;base64,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)
Amaya made 10 three point shots out of 25 attempts. If she then goes on to make consecutive three point shots, her success would be given by the function![f left parenthesis x right parenthesis equals fraction numerator x plus 10 over denominator x plus 25 end fraction](data:image/png;base64,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)
Maths-General
Maths-
Which function has a graph with a horizontal asymptote at y = -1.
![text a. end text straight F left parenthesis straight x right parenthesis equals fraction numerator x plus 5 over denominator x minus 3 end fraction](data:image/png;base64,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)
Which function has a graph with a horizontal asymptote at y = -1.
![text a. end text straight F left parenthesis straight x right parenthesis equals fraction numerator x plus 5 over denominator x minus 3 end fraction](data:image/png;base64,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)
Maths-General
Maths-
Graph the function, labelling all horizontal or vertical asymptotes of the form x =a or y = b
![straight F left parenthesis straight x right parenthesis equals fraction numerator 6 x squared minus 12 x over denominator x squared plus 5 x minus 24 end fraction](data:image/png;base64,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)
Graph the function, labelling all horizontal or vertical asymptotes of the form x =a or y = b
![straight F left parenthesis straight x right parenthesis equals fraction numerator 6 x squared minus 12 x over denominator x squared plus 5 x minus 24 end fraction](data:image/png;base64,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)
Maths-General
Maths-
Graph the function, labelling all horizontal or vertical asymptotes of the form x = a or y = b
![F left parenthesis x right parenthesis equals fraction numerator x squared plus x minus 2 over denominator 2 x squared minus 9 x minus 18 end fraction](data:image/png;base64,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)
Graph the function, labelling all horizontal or vertical asymptotes of the form x = a or y = b
![F left parenthesis x right parenthesis equals fraction numerator x squared plus x minus 2 over denominator 2 x squared minus 9 x minus 18 end fraction](data:image/png;base64,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)
Maths-General