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Thread: 4 college algebra questions

  1. #1
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    4 college algebra questions

    1. a farmer has 1442 meters of fencing available to enclose a rectangular portion of his land. One side of the rectangle being fenced lies along a river, so only three sides require fencing.
    (a) express the area A of the rectangle as a function of x, where x is the
    length of the side parallel to the river.
    (b) for what value of x is the area largest

    2. Use synthetic division to complete the indicated factorization.
    x4 -2x3-25x2+26x+120=(x-3)(x+2)( )
    A: x2-x-19 B: x2+x+20 C: x2-x-20 D: x2-2x-19

    3. Given that one zero of P(x)=x3+7x2-23x-185 is 6-i, which of the following is also a zero of P(x)?
    A: 6-I B: -6+I C: 1-6i D: -1-6i

    4. Find the complete factorization of 2x4+3x3-12x2-7x+6 if (2x-1) is one of the factors
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  2. #2
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    Quote Originally Posted by Lane
    1. a farmer has 1442 meters of fencing available to enclose a rectangular portion of his land. One side of the rectangle being fenced lies along a river, so only three sides require fencing.
    (a) express the area A of the rectangle as a function of x, where x is the
    length of the side parallel to the river.
    (b) for what value of x is the area largest
    (a) we now the area of a rectangle is length times width, and we know the length is $\displaystyle x$ we also know that both the widths added together are $\displaystyle 1442-x$ since there are 2 sides for width we divide what they are added together by 2. $\displaystyle w=\frac{1442-x}{2}$.

    so now we take what we know...

    $\displaystyle x=\text{length}$
    $\displaystyle w=\frac{1442-x}{2}$
    and the Area is the function of $\displaystyle x$

    and solve for area...
    $\displaystyle l\cdot w=A$

    $\displaystyle x\cdot\left(\frac{1442-x}{2}\right)=f(x)$

    $\displaystyle \frac{1442x-x^2}{2}=f(x)$

    $\displaystyle f(x)=\frac{-x^2}{2}+\frac{1442x}{2}$

    $\displaystyle f(x)=\frac{-1}{2}x^2+721x$ now we are left with a quadratic equation that opens down, so we find the vertex...

    $\displaystyle \text{vertex}=\frac{-b}{2a}$

    $\displaystyle \frac{-721}{2\cdot\frac{-1}{2}}$

    $\displaystyle \frac{-721}{-1}$

    $\displaystyle 721$

    so the area is largest when $\displaystyle x=721$
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  3. #3
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    Quote Originally Posted by Lane

    3. Given that one zero of P(x)=x3+7x2-23x-185 is 6-i, which of the following is also a zero of P(x)?
    A: 6-I B: -6+I C: 1-6i D: -1-6i
    Since, $\displaystyle 6-i$ is a solution then $\displaystyle 6+i$ is a solution cuz, solutions appear in conjegate paris.
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  4. #4
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    Hello, Lane!

    2. Use synthetic division to complete the indicated factorization.
    . . . $\displaystyle 2x^3-25x^2+26x+120 \;= \;(x-3)(x+2)(\qquad)$

    . . . $\displaystyle A)\;x^2-x-19\qquad B)\; x^2+x+20\qquad C)$$\displaystyle x^2-x-20\qquad D)\;x^2-2x-19$

    Do you how to do synthetic division?
    Code:
    
          3  |   1  -2   -25   26   120 
             |       3     3  -66  -120
             | - - - - - - - - - - - - -
                 1   1   -22  -40     0
    
         -2  |   1   1   -22  -40
             |      -2     2   40
             | - - - - - - - - - -
                 1  -1   -20    0

    The third factor is: .$\displaystyle x^2 - x - 20$ . . . Answer (C)



    4. Find the complete factorization of $\displaystyle 2x^4+3x^3-12x^2-7x+6$
    if $\displaystyle (2x-1)$ is one of the factors.

    Using long division or synthetic division, we find that:
    . . $\displaystyle x^4 - 2x^3 - 25x^2 + 26x + 120 \;=\;(2x - 1)(x^3 +2 x^2 - 5x - 6)$

    From the Rational Roots theorem, we test: .$\displaystyle x\:=\:\pm1,\;\pm2,\;\pm3$
    . . and find that: .$\displaystyle x\:=\:-1,\;2,\;-3$ are zeros of the cubic.

    Therefore, the factorization is: .$\displaystyle (2x - 1)(x + 1)(x - 2)(x + 3)$

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