Chapter 15

Use the factor theorem to determine whether or not the second expression is a factor of the first expression. Do not use synthetic division. $$x^{7}-128^{-1}, x+2^{-1}$$

Use a calculator to solve the given equations to the nearest 0.01. $$8 x^{4}+36 x^{3}+35 x^{2}-4 x-4=0$$

Perform the indicated divisions by synthetic division. $$\left(x^{3}+2 x^{2}-x-2\right) \div(x-1)$$

Perform the indicated divisions by synthetic division. $$\left(x^{3}-3 x^{2}-x+2\right) \div(x-3)$$

Find the remaining roots of the given equations using synthetic division, given the roots indicated. \(x^{6}+2 x^{5}-4 x^{4}-10 x^{3}-41 x^{2}-72 x-36=0\) \((-1 \text { is a double root; } 2 j\) is a root)

Solve the given problems. Use a calculator to solve if necessary. Solve the following system algebraically: \(y=x^{4}-11 x^{2} ; \quad y=12 x-4\)

Perform the indicated divisions by synthetic division. $$\left(x^{3}+2 x^{2}-3 x+4\right) \div(x+4)$$

Solve the given problems. Use a calculator to solve if necessary. Find rational values of \(a\) such that \((x-a)\) will divide into \(x^{3}+x^{2}-4 x-4\) with a remainder of zero.

Perform the indicated divisions by synthetic division. $$\left(2 x^{3}-4 x^{2}+x-1\right) \div(x+2)$$

Solve the given problems. Use a calculator to solve if necessary. Where does the graph of the function \(f(x)=4 x^{3}+3 x^{2}-20 x-15\) cross the \(x\) -axis?

Perform the indicated divisions by synthetic division. $$\left(p^{6}-6 p^{3}-2 p^{2}-6\right) \div(p-2)$$

Solve the given problems. Use a calculator to solve if necessary. Where does the graph of the function \(f(s)=2 s^{4}-s^{3}-5 s^{2}+7 s-6\) cross the \(s\) -axis?

Perform the indicated divisions by synthetic division. $$\left(x^{5}+4 x^{4}-8\right) \div(x+1)$$

Solve the given problems. Find \(k\) such that \(x-2\) is a factor of \(2 x^{3}+k x^{2}-k x-2\).

Solve the given problems. Use a calculator to solve if necessary. By checking only the equation and the coefficients, determine the smallest and largest possible rational roots of the equation \(2 x^{4}+x^{2}-22 x+8=0\).

Perform the indicated divisions by synthetic division. $$\left(x^{7}-128\right) \div(x-2)$$

Solve the given problems. Find \(k\) such that \(x-1\) is a factor of \(x^{3}-4 x^{2}-k x+2\).

Solve the given problems. Use a calculator to solve if necessary. By checking only the equation and the coefficients, determine the smallest and largest possible rational roots of the equation \(2 x^{4}+x^{2}+22 x+26=0\).

Solve the given problems. Equations of the form \(y^{2}=x^{3}+a x+b\) are called elliptic curves and are used in cryptography. If \(y=3\) for the curve \(y^{2}=x^{3}-4 x+6,\) use synthetic division to show that one possible value of \(x\) is \(x=-1\). Then find any other possible values of \(x\).

Solve the given problems. Use a calculator to solve if necessary. The angular acceleration \(\alpha\) (in \(\mathrm{rad} / \mathrm{s}^{2}\) ) of the wheel of a car is given by \(\alpha=-0.2 t^{3}+t^{2},\) where \(t\) is the time (in s). For what values of \(t\) is \(\alpha=2.0 \mathrm{rad} / \mathrm{s}^{2} ?\)

Perform the indicated divisions by synthetic division. $$\left(2 x^{4}+x^{3}+3 x^{2}-1\right) \div(2 x-1)$$

Solve the given problems. Form a polynomial equation of the smallest possible degree and with integer coefficients, having a double root of \(3,\) and a root of \(j\).

Solve the given problems. Use a calculator to solve if necessary. In finding one of the dimensions \(d\) (in in.) of the support columns of a building, the equation \(3 d^{3}+5 d^{2}-400 d-18,000=0\) is found. What is this dimension?

Perform the indicated divisions by synthetic division. $$\left(6 t^{4}+5 t^{3}-10 t+4\right) \div(3 t-2)$$

Solve the given problems. Use a calculator to solve if necessary. The deflection \(y\) of a beam at a horizontal distance \(x\) from one end is given by \(y=k\left(x^{4}-2 L x^{3}-L^{3} x\right),\) where \(L\) is the length of the beam and \(k\) is a constant. For what values of \(x\) is the deflection zero?

Use the factor theorem and synthetic division to determine whether or not the second expression is a factor of the first. $$2 x^{5}-x^{3}+3 x^{2}-4 ; \quad x+1$$

Solve the given problems. Use a calculator to solve if necessary. The specific gravity \(s\) of a sphere of radius \(r\) that sinks to a depth \(h\) in water is given by \(s=\frac{3 r h^{2}-h^{3}}{4 r^{3}} .\) Find the depth to which a spherical buoy of radius \(4.0 \mathrm{cm}\) sinks if \(s=0.50\).

Use the factor theorem and synthetic division to determine whether or not the second expression is a factor of the first. $$t^{5}-3 t^{4}-t^{2}-6 ; t-3$$

Use the factor theorem and synthetic division to determine whether or not the second expression is a factor of the first. $$4 x^{3}-9 x^{2}+2 x-2 ; \quad x-\frac{1}{4}$$

Solve the given problems. Use a calculator to solve if necessary. The pressure difference \(p\) (in \(\mathrm{kPa}\) ) at a distance \(x\) (in \(\mathrm{km}\) ) from one end of an oil pipeline is given by \(p=x^{5}-3 x^{4}-x^{2}+7 x\). If the pipeline is \(4 \mathrm{km}\) long, where is \(p=0 ?\)

Use the factor theorem and synthetic division to determine whether or not the second expression is a factor of the first. $$3 x^{3}-5 x^{2}+x+1 ; \quad x+\frac{1}{3}$$

Solve the given problems. Use a calculator to solve if necessary. A rectangular tray is made from a square piece of sheet metal \(10.0 \mathrm{cm}\) on a side by cutting equal squares from each corner, bending up the sides, and then welding them together. How long is the side of the square that must be cut out if the volume of the tray is \(70.0 \mathrm{cm}^{3} ?\)

Use the factor theorem and synthetic division to determine whether or not the second expression is a factor of the first. $$2 Z^{4}-Z^{3}-4 Z^{2}+1 ; \quad 2 Z-1$$

Solve the given problems. Use a calculator to solve if necessary. The angle \(\theta\) (in degrees) of a robot arm with the horizontal as a function of time \(t\) (in s) is given by \(\theta=15+20 t^{2}-4 t^{3}\) for \(0 \leq t \leq 5\) s. Find \(t\) for \(\theta=40^{\circ}\).

Use the factor theorem and synthetic division to determine whether or not the second expression is a factor of the first. $$6 x^{4}+5 x^{3}-x^{2}+6 x-2 ; \quad 3 x-1$$

Solve the given problems. Use a calculator to solve if necessary. The radii of four different-sized ball bearings differ by \(1.00 \mathrm{mm}\) in radius from one size to the next. If the volume of the largest equals the volumes of the other three combined, find the radii.

Use the factor theorem and synthetic division to determine whether or not the second expression is a factor of the first. $$4 x^{4}+2 x^{3}-8 x^{2}+3 x+12 ; \quad 2 x+3$$

Solve the given problems. Use a calculator to solve if necessary. A rectangular safe is to be made of steel of uniform thickness, including the door. The inside dimensions are \(1.20 \mathrm{m}, 1.20 \mathrm{m},\) and 2.00 \(\mathrm{m}\). If the volume of steel is \(1.25 \mathrm{m}^{3}\), find its thickness.

Use the factor theorem and synthetic division to determine whether or not the second expression is a factor of the first. $$3 x^{4}-2 x^{3}+x^{2}+15 x+4 ; \quad 3 x+4$$

Use synthetic division to determine whether or not the given numbers are zeros of the given functions. $$x^{4}-5 x^{3}-15 x^{2}+5 x+14 ; 7$$

Solve the given problems. Use a calculator to solve if necessary. Each of three revolving doors has a perimeter of \(6.60 \mathrm{m}\) and revolves through a volume of \(9.50 \mathrm{m}^{3}\) in one revolution about their common vertical side. What are the door's dimensions?

Use synthetic division to determine whether or not the given numbers are zeros of the given functions. $$r^{4}+5 r^{3}-18 r-8 ; \quad-4$$

Use synthetic division to determine whether or not the given numbers are zeros of the given functions. $$85 x^{3}+348 x^{2}-263 x+120 ; \quad-4.8$$

Solve the given problems. Use a calculator to solve if necessary. An equation \(f(x)=0\) involves only odd powers of \(x\) with positive coefficients. Explain why this equation has no real root except \(x=0\).

Use synthetic division to determine whether or not the given numbers are zeros of the given functions. $$2 x^{3}+13 x^{2}+10 x-4 ; \frac{1}{2}$$

Solve the given problems. If \(f(x)=2 x^{3}+3 x^{2}-19 x-4,\) and \(f(x)=(x+4) g(x)\) find \(g(x)\)

Solve the given problems. By division, show that \(x^{2}+2\) is a factor of \(f(x)=3 x^{3}-x^{2}+6 x-2 .\) May we therefore conclude that \(f(-2)=0 ?\) Explain.

Solve the given problems. For what value of \(k\) is \(x-2\) a factor of \(f(x)=2 x^{3}+k x^{2}-x+14 ?\)

Solve the given problems. For what value of \(k\) is \(x+1\) a factor of \(f(x)=3 x^{4}+3 x^{3}+2 x^{2}+k x-4 ?\)

Solve the given problems. If \(f(x)=-g(x),\) do the functions have the same zeros? Explain.