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The dev project

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Matt Beck
Matt Beck

This document contains 4 math problems with step-by-step solutions: 1) factoring and graphing a polynomial function, finding vertical and horizontal asymptotes, x-intercepts, and holes; 2) finding the maximum area of a garden within a fence of a given length; 3) fully distributing a polynomial; 4) factoring a quadratic equation.

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By Matt
Problem 1 Graph: x³+9x²-64x-576 x³+7x²-50x-336 Given: x=-6
To solve this problem you have to get the numerator and denominator to simplest forms Since there are four proportional terms we can group the numerator so we start of by factoring out the GCF After doing that we end up with this because both sides are multiplied by (x+9) Because of difference of squares we are able to simplify it even more
Since the denominator is not proportional we must long divide using -6(given) To long divide you must find out what you have to multiply by x to get x³( its x²) then multiply that by the rest of the polynomial then subtract the new from the old and repeat We come out with this Reduced to
Now we must get what we need for the graph out of the equations Factored Standard Vertical Asymptote: -6,7 in denominator of factor form when we put 0 in for x Horizontal Asymptote: 1 in standard form taking highest power then dividing X int: -9,8 in numerator in factor form when we put 0 in for x Y int: 12/7 taking to lowest powers from standard form and dividing Hole: -8 in factored form when numerator and denominator are the same
The Graph
Question 2 • Mark has 200 feet of fence he want to make a fence around his garden and use his house as one of the sides to make the garden bigger. What is the maximum area Mark can have? • Find domain and range when done
We must start of by making 2 equations that show perimeter and area Since we have 200 feet of fence and we have 3 sides 2 of which are the same the equation is 200=2x+y
We must solve for a now Start of by solving the Perimeter formula 200-2x=y Put this equation in for y in the area equation A=x(200-2x) A= -2x²+200x
To find the maximum value use the formula -b/2a to find the x of the vertex -200/-4= 50 Plug x into the equation to get the maximum -2(50)²+200(50)=5000 D:(-∞,∞) No number x cannot equal R:(-∞,5000] No minimum value and already found the maximum
Question 3 Distribute (x+4)(x-4)(x+6)(x-10)(x+2)
Start of by distributing (x+4) to (x-4) Then you should get Which simplifies to
Distribute (x²-16) to (x+6) You should get
Distribute (x³+6x²-16x-96) to (x-10) You should come out with Simplify to
Distribute (x⁴-4x³-76x²-256x+960) to (x+2) Comes out to Simplifies to
Question 4 Factor 10x²+82x+27=x²-2x
Get it Equal to Zero Subtract x² then add -2x Which should get you 9x²+84x+27=0
Set it up (9x+ )( + ) You must then find to numbers that multiply to get 27 but add to get 84x
You should come up with (9x+3)(x+9) X=3/9 X=9

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The dev project

  • 3. To solve this problem you have to get the numerator and denominator to simplest forms Since there are four proportional terms we can group the numerator so we start of by factoring out the GCF After doing that we end up with this because both sides are multiplied by (x+9) Because of difference of squares we are able to simplify it even more
  • 4. Since the denominator is not proportional we must long divide using -6(given) To long divide you must find out what you have to multiply by x to get x³( its x²) then multiply that by the rest of the polynomial then subtract the new from the old and repeat We come out with this Reduced to
  • 5. Now we must get what we need for the graph out of the equations Factored Standard Vertical Asymptote: -6,7 in denominator of factor form when we put 0 in for x Horizontal Asymptote: 1 in standard form taking highest power then dividing X int: -9,8 in numerator in factor form when we put 0 in for x Y int: 12/7 taking to lowest powers from standard form and dividing Hole: -8 in factored form when numerator and denominator are the same
  • 7. Question 2 • Mark has 200 feet of fence he want to make a fence around his garden and use his house as one of the sides to make the garden bigger. What is the maximum area Mark can have? • Find domain and range when done
  • 8. We must start of by making 2 equations that show perimeter and area Since we have 200 feet of fence and we have 3 sides 2 of which are the same the equation is 200=2x+y
  • 9. We must solve for a now Start of by solving the Perimeter formula 200-2x=y Put this equation in for y in the area equation A=x(200-2x) A= -2x²+200x
  • 10. To find the maximum value use the formula -b/2a to find the x of the vertex -200/-4= 50 Plug x into the equation to get the maximum -2(50)²+200(50)=5000 D:(-∞,∞) No number x cannot equal R:(-∞,5000] No minimum value and already found the maximum
  • 12. Start of by distributing (x+4) to (x-4) Then you should get Which simplifies to
  • 13. Distribute (x²-16) to (x+6) You should get
  • 14. Distribute (x³+6x²-16x-96) to (x-10) You should come out with Simplify to
  • 15. Distribute (x⁴-4x³-76x²-256x+960) to (x+2) Comes out to Simplifies to
  • 17. Get it Equal to Zero Subtract x² then add -2x Which should get you 9x²+84x+27=0
  • 18. Set it up (9x+ )( + ) You must then find to numbers that multiply to get 27 but add to get 84x
  • 19. You should come up with (9x+3)(x+9) X=3/9 X=9