List all of the possible rational roots of the following functions:
1) \(f(x)=2x^4-x^3-8x^2+x+6\)
2) \(g(x)=3x^4-20x^3-22x^2+4x+3\)
3) \(h(x)=6x^3+19x^2-52x+15\)
4) Given the quartic function \(j(x)=x^4-3x^3-2x^2-x-3\), Alex thinks that \(\pm1\) and \(\pm3\) are the four solutions to \(j(x)\). Julia disagrees and says that \(j(x)\) only has one rational zero at \(x=-1\). Who is correct and why?
For questions #5-7, use synthetic or long division to determine if the given x-value is a zero of the function.
5) \(k(x)=2x^3+7x^2-x-12;\ \ \ x=-6\)
6) \(m(x)=3x^4-20x^3-7x^2+2x-14;\ \ \ x=7\)
7) \(n(x)=2x^3-x^2-16x+15;\ \ \ x=\large\frac{5}{2}\)
Solve the following equations:
8) \(0=x^4+3x^2-40\)
9) \(x^3+27=0\)
10) \(x^4-16=0\)
11) \(2x^5+6x^3-36x=0\)
12) \(5x^4+5x^3-30x^2=0\)
13) \(9x^2+25=0\)
14) \(0=4x^5-36x\)
15) \(x^4-5x^2=14\)
16) \(8x^3=64\)
For questions #17-20, find all zeros of the following polynomial functions.
You may use a graphing device to graph the function and find a real zero. Then use synthetic division with
that zero until you get it to an expression that you can factor, complete the square, or use quadratic formula.
17) \(p(x)=x^3-2x^2-13x+6\)
18) \(q(x)=x^3+5x^2-2x-4\)
19) \(r(x)=x^4-2x^3+2x^2-10x-15\)
20) \(s(x)=x^4+x^3-17x^2-18x+24\)
For questions #21-24, solve over the complex numbers.
You may use a graphing device to graph the function and find a real zero. Then use synthetic division with
that zero until you get it to an expression that you can factor, complete the square, or use quadratic formula.
21) \(0=2x^3+x^2-25x+12\)
22) \(0=x^3-x^2-32x-10\)
23) \(0=x^4+2x^3+6x^2+18x-27\)
24) \(0=x^5+2x^4-4x^3-8x^2-12x-24\)
25) Given that \(x=-3\) is a root of \(t(x)=5x^3+12x^2-8x+3\), find the remaining roots.
26) Given that \(x=\large-\frac{2}{3}\) is a root of \(u(x)=6x^3-14x^2+3x+10\), find the remaining roots.
27) Given that \(x=\large-\frac{5}{2}\) is a root of \(v(x)=2x^3-15x^2-44x+15\), find the remaining roots.
Review
28) Simplify: \(\left(\Large\frac{4x^2y^{-3}z^{6}}{6x^{-5}y^0z^{11}}\right)^3\small\)
29) Simplify: \(\large{x\strut{^{-4}}}\Large\cdot\frac{1}{x^2}\cdot\frac{1}{x^{-5}}\)
30) Given \(w(x)=-2(x+2)^2(x-1)(x+1)\), identify the degree, leading coefficient, end behavior, and x- and y-intercepts. Then use this information to sketch a graph of the polynomial.
Solution Bank
1) \(f(x)=2x^4-x^3-8x^2+x+6\)
2) \(g(x)=3x^4-20x^3-22x^2+4x+3\)
3) \(h(x)=6x^3+19x^2-52x+15\)
4) Given the quartic function \(j(x)=x^4-3x^3-2x^2-x-3\), Alex thinks that \(\pm1\) and \(\pm3\) are the four solutions to \(j(x)\). Julia disagrees and says that \(j(x)\) only has one rational zero at \(x=-1\). Who is correct and why?
For questions #5-7, use synthetic or long division to determine if the given x-value is a zero of the function.
5) \(k(x)=2x^3+7x^2-x-12;\ \ \ x=-6\)
6) \(m(x)=3x^4-20x^3-7x^2+2x-14;\ \ \ x=7\)
7) \(n(x)=2x^3-x^2-16x+15;\ \ \ x=\large\frac{5}{2}\)
Solve the following equations:
8) \(0=x^4+3x^2-40\)
9) \(x^3+27=0\)
10) \(x^4-16=0\)
11) \(2x^5+6x^3-36x=0\)
12) \(5x^4+5x^3-30x^2=0\)
13) \(9x^2+25=0\)
14) \(0=4x^5-36x\)
15) \(x^4-5x^2=14\)
16) \(8x^3=64\)
For questions #17-20, find all zeros of the following polynomial functions.
You may use a graphing device to graph the function and find a real zero. Then use synthetic division with
that zero until you get it to an expression that you can factor, complete the square, or use quadratic formula.
17) \(p(x)=x^3-2x^2-13x+6\)
18) \(q(x)=x^3+5x^2-2x-4\)
19) \(r(x)=x^4-2x^3+2x^2-10x-15\)
20) \(s(x)=x^4+x^3-17x^2-18x+24\)
For questions #21-24, solve over the complex numbers.
You may use a graphing device to graph the function and find a real zero. Then use synthetic division with
that zero until you get it to an expression that you can factor, complete the square, or use quadratic formula.
21) \(0=2x^3+x^2-25x+12\)
22) \(0=x^3-x^2-32x-10\)
23) \(0=x^4+2x^3+6x^2+18x-27\)
24) \(0=x^5+2x^4-4x^3-8x^2-12x-24\)
25) Given that \(x=-3\) is a root of \(t(x)=5x^3+12x^2-8x+3\), find the remaining roots.
26) Given that \(x=\large-\frac{2}{3}\) is a root of \(u(x)=6x^3-14x^2+3x+10\), find the remaining roots.
27) Given that \(x=\large-\frac{5}{2}\) is a root of \(v(x)=2x^3-15x^2-44x+15\), find the remaining roots.
Review
28) Simplify: \(\left(\Large\frac{4x^2y^{-3}z^{6}}{6x^{-5}y^0z^{11}}\right)^3\small\)
29) Simplify: \(\large{x\strut{^{-4}}}\Large\cdot\frac{1}{x^2}\cdot\frac{1}{x^{-5}}\)
30) Given \(w(x)=-2(x+2)^2(x-1)(x+1)\), identify the degree, leading coefficient, end behavior, and x- and y-intercepts. Then use this information to sketch a graph of the polynomial.
Solution Bank