Practice Problems
For each polynomial function, determine the following without graphing:
(a) Degree of the polynomial
(b) Leading coefficient
(c) End behavior as x approaches positive and negative infinity
(d) Maximum possible number of real zeros
(e) Maximum possible number of turning points
(f) Actual real zeros of the polynomial
(g) Whether the graph crosses or touches the x-axis at each zero
\(\textbf{1)}\) \(f(x)=x^4+2x^3-4x+2\)(a) Degree: \(4\)
(b) Leading coefficient: \(1\)
(c) End behavior:
\(x\to-\infty,\quad f(x)\to\infty\)
\(x\to\infty,\quad f(x)\to\infty\)
(d) Maximum real zeros: \(4\)
(e) Maximum turning points: \(3\)
(f) Actual real zeros: None.
(g) Crosses or touches: Neither. The graph remains above the x-axis.
(b) Leading coefficient: \(1\)
(c) End behavior:
\(x\to-\infty,\quad f(x)\to\infty\)
\(x\to\infty,\quad f(x)\to\infty\)
(d) Maximum real zeros: \(4\)
(e) Maximum turning points: \(3\)
(f) Actual real zeros: None.
(g) Crosses or touches: Neither. The graph remains above the x-axis.
\(\textbf{2)}\) \(f(x)=6x^3+x^2-12\)(a) Degree: \(3\)
(b) Leading coefficient: \(6\)
(c) End behavior:
\(x\to-\infty,\quad f(x)\to-\infty\)
\(x\to\infty,\quad f(x)\to\infty\)
(d) Maximum real zeros: \(3\)
(e) Maximum turning points: \(2\)
(f) Actual real zeros: \(x\approx1.2067\)
(g) Crosses or touches: Crosses at its single real zero.
(b) Leading coefficient: \(6\)
(c) End behavior:
\(x\to-\infty,\quad f(x)\to-\infty\)
\(x\to\infty,\quad f(x)\to\infty\)
(d) Maximum real zeros: \(3\)
(e) Maximum turning points: \(2\)
(f) Actual real zeros: \(x\approx1.2067\)
(g) Crosses or touches: Crosses at its single real zero.
\(\textbf{3)}\) \(f(x)=-2x^2+3\)(a) Degree: \(2\)
(b) Leading coefficient: \(-2\)
(c) End behavior:
\(x\to-\infty,\quad f(x)\to-\infty\)
\(x\to\infty,\quad f(x)\to-\infty\)
(d) Maximum real zeros: \(2\)
(e) Maximum turning points: \(1\)
(f) Actual real zeros: \(x=\pm\frac{\sqrt6}{2}\)
(g) Crosses or touches: Crosses at both zeros, each with multiplicity 1.
(b) Leading coefficient: \(-2\)
(c) End behavior:
\(x\to-\infty,\quad f(x)\to-\infty\)
\(x\to\infty,\quad f(x)\to-\infty\)
(d) Maximum real zeros: \(2\)
(e) Maximum turning points: \(1\)
(f) Actual real zeros: \(x=\pm\frac{\sqrt6}{2}\)
(g) Crosses or touches: Crosses at both zeros, each with multiplicity 1.
\(\textbf{4)}\) \(f(x)=-8x^2-3x+5\)(a) Degree: \(2\)
(b) Leading coefficient: \(-8\)
(c) End behavior:
\(x\to-\infty,\quad f(x)\to-\infty\)
\(x\to\infty,\quad f(x)\to-\infty\)
(d) Maximum real zeros: \(2\)
(e) Maximum turning points: \(1\)
(f) Actual real zeros: \(x=-1,\quad x=\frac58\)
(g) Crosses or touches: Crosses at both zeros, each with multiplicity 1.
(b) Leading coefficient: \(-8\)
(c) End behavior:
\(x\to-\infty,\quad f(x)\to-\infty\)
\(x\to\infty,\quad f(x)\to-\infty\)
(d) Maximum real zeros: \(2\)
(e) Maximum turning points: \(1\)
(f) Actual real zeros: \(x=-1,\quad x=\frac58\)
(g) Crosses or touches: Crosses at both zeros, each with multiplicity 1.
\(\textbf{5)}\) \(f(x)=4x\)(a) Degree: \(1\)
(b) Leading coefficient: \(4\)
(c) End behavior:
\(x\to-\infty,\quad f(x)\to-\infty\)
\(x\to\infty,\quad f(x)\to\infty\)
(d) Maximum real zeros: \(1\)
(e) Maximum turning points: \(0\)
(f) Actual real zeros: \(x=0\)
(g) Crosses or touches: Crosses at \(x=0\), multiplicity 1.
(b) Leading coefficient: \(4\)
(c) End behavior:
\(x\to-\infty,\quad f(x)\to-\infty\)
\(x\to\infty,\quad f(x)\to\infty\)
(d) Maximum real zeros: \(1\)
(e) Maximum turning points: \(0\)
(f) Actual real zeros: \(x=0\)
(g) Crosses or touches: Crosses at \(x=0\), multiplicity 1.
\(\textbf{6)}\) \(f(x)=-x\)(a) Degree: \(1\)
(b) Leading coefficient: \(-1\)
(c) End behavior:
\(x\to-\infty,\quad f(x)\to\infty\)
\(x\to\infty,\quad f(x)\to-\infty\)
(d) Maximum real zeros: \(1\)
(e) Maximum turning points: \(0\)
(f) Actual real zeros: \(x=0\)
(g) Crosses or touches: Crosses at \(x=0\), multiplicity 1.
(b) Leading coefficient: \(-1\)
(c) End behavior:
\(x\to-\infty,\quad f(x)\to\infty\)
\(x\to\infty,\quad f(x)\to-\infty\)
(d) Maximum real zeros: \(1\)
(e) Maximum turning points: \(0\)
(f) Actual real zeros: \(x=0\)
(g) Crosses or touches: Crosses at \(x=0\), multiplicity 1.
\(\textbf{7)}\) \(f(x)=-4x^3-20x^2+3x-8\)(a) Degree: \(3\)
(b) Leading coefficient: \(-4\)
(c) End behavior:
\(x\to-\infty,\quad f(x)\to\infty\)
\(x\to\infty,\quad f(x)\to-\infty\)
(d) Maximum real zeros: \(3\)
(e) Maximum turning points: \(2\)
(f) Actual real zeros: \(x\approx-5.2172\)
(g) Crosses or touches: Crosses at its single real zero.
(b) Leading coefficient: \(-4\)
(c) End behavior:
\(x\to-\infty,\quad f(x)\to\infty\)
\(x\to\infty,\quad f(x)\to-\infty\)
(d) Maximum real zeros: \(3\)
(e) Maximum turning points: \(2\)
(f) Actual real zeros: \(x\approx-5.2172\)
(g) Crosses or touches: Crosses at its single real zero.
\(\textbf{8)}\) \(f(x)=x^4-2x^2\)(a) Degree: \(4\)
(b) Leading coefficient: \(1\)
(c) End behavior:
\(x\to-\infty,\quad f(x)\to\infty\)
\(x\to\infty,\quad f(x)\to\infty\)
(d) Maximum real zeros: \(4\)
(e) Maximum turning points: \(3\)
(f) Actual real zeros: \(x=-\sqrt2,\quad x=0,\quad x=\sqrt2\)
(g) Crosses or touches:
\(x=0\): Touches (multiplicity 2).
\(x=\pm\sqrt2\): Crosses (multiplicity 1).
(b) Leading coefficient: \(1\)
(c) End behavior:
\(x\to-\infty,\quad f(x)\to\infty\)
\(x\to\infty,\quad f(x)\to\infty\)
(d) Maximum real zeros: \(4\)
(e) Maximum turning points: \(3\)
(f) Actual real zeros: \(x=-\sqrt2,\quad x=0,\quad x=\sqrt2\)
(g) Crosses or touches:
\(x=0\): Touches (multiplicity 2).
\(x=\pm\sqrt2\): Crosses (multiplicity 1).
\(\textbf{9)}\) \(f(x)=-x^4+2x^3+4x^2-8x\)(a) Degree: \(4\)
(b) Leading coefficient: \(-1\)
(c) End behavior:
\(x\to-\infty,\quad f(x)\to-\infty\)
\(x\to\infty,\quad f(x)\to-\infty\)
(d) Maximum real zeros: \(4\)
(e) Maximum turning points: \(3\)
(f) Actual real zeros: \(x=-2,\quad x=0,\quad x=2\)
(g) Crosses or touches:
\(x=-2\): Crosses (multiplicity 1).
\(x=0\): Crosses (multiplicity 1).
\(x=2\): Touches (multiplicity 2).
(b) Leading coefficient: \(-1\)
(c) End behavior:
\(x\to-\infty,\quad f(x)\to-\infty\)
\(x\to\infty,\quad f(x)\to-\infty\)
(d) Maximum real zeros: \(4\)
(e) Maximum turning points: \(3\)
(f) Actual real zeros: \(x=-2,\quad x=0,\quad x=2\)
(g) Crosses or touches:
\(x=-2\): Crosses (multiplicity 1).
\(x=0\): Crosses (multiplicity 1).
\(x=2\): Touches (multiplicity 2).
\(\textbf{10)}\) \(f(x)=x^3-4x\)(a) Degree: \(3\)
(b) Leading coefficient: \(1\)
(c) End behavior:
\(x\to-\infty,\quad f(x)\to-\infty\)
\(x\to\infty,\quad f(x)\to\infty\)
(d) Maximum real zeros: \(3\)
(e) Maximum turning points: \(2\)
(f) Actual real zeros: \(x=-2,\quad x=0,\quad x=2\)
(g) Crosses or touches: Crosses at all three zeros (multiplicity 1 each).
(b) Leading coefficient: \(1\)
(c) End behavior:
\(x\to-\infty,\quad f(x)\to-\infty\)
\(x\to\infty,\quad f(x)\to\infty\)
(d) Maximum real zeros: \(3\)
(e) Maximum turning points: \(2\)
(f) Actual real zeros: \(x=-2,\quad x=0,\quad x=2\)
(g) Crosses or touches: Crosses at all three zeros (multiplicity 1 each).
\(\textbf{11)}\) \(f(x)=-x^3+9x\)(a) Degree: \(3\)
(b) Leading coefficient: \(-1\)
(c) End behavior:
\(x\to-\infty,\quad f(x)\to\infty\)
\(x\to\infty,\quad f(x)\to-\infty\)
(d) Maximum real zeros: \(3\)
(e) Maximum turning points: \(2\)
(f) Actual real zeros: \(x=-3,\quad x=0,\quad x=3\)
(g) Crosses or touches: Crosses at all three zeros (multiplicity 1 each).
(b) Leading coefficient: \(-1\)
(c) End behavior:
\(x\to-\infty,\quad f(x)\to\infty\)
\(x\to\infty,\quad f(x)\to-\infty\)
(d) Maximum real zeros: \(3\)
(e) Maximum turning points: \(2\)
(f) Actual real zeros: \(x=-3,\quad x=0,\quad x=3\)
(g) Crosses or touches: Crosses at all three zeros (multiplicity 1 each).
\(\textbf{12)}\) \(f(x)=(x-1)^2(x+3)\)(a) Degree: \(3\)
(b) Leading coefficient: \(1\)
(c) End behavior:
\(x\to-\infty,\quad f(x)\to-\infty\)
\(x\to\infty,\quad f(x)\to\infty\)
(d) Maximum real zeros: \(3\)
(e) Maximum turning points: \(2\)
(f) Actual real zeros: \(x=-3,\quad x=1\)
(g) Crosses or touches:
\(x=-3\): Crosses (multiplicity 1).
\(x=1\): Touches (multiplicity 2).
(b) Leading coefficient: \(1\)
(c) End behavior:
\(x\to-\infty,\quad f(x)\to-\infty\)
\(x\to\infty,\quad f(x)\to\infty\)
(d) Maximum real zeros: \(3\)
(e) Maximum turning points: \(2\)
(f) Actual real zeros: \(x=-3,\quad x=1\)
(g) Crosses or touches:
\(x=-3\): Crosses (multiplicity 1).
\(x=1\): Touches (multiplicity 2).
\(\textbf{13)}\) \(f(x)=-(x+2)^2(x-4)\)(a) Degree: \(3\)
(b) Leading coefficient: \(-1\)
(c) End behavior:
\(x\to-\infty,\quad f(x)\to\infty\)
\(x\to\infty,\quad f(x)\to-\infty\)
(d) Maximum real zeros: \(3\)
(e) Maximum turning points: \(2\)
(f) Actual real zeros: \(x=-2,\quad x=4\)
(g) Crosses or touches:
\(x=-2\): Touches (multiplicity 2).
\(x=4\): Crosses (multiplicity 1).
(b) Leading coefficient: \(-1\)
(c) End behavior:
\(x\to-\infty,\quad f(x)\to\infty\)
\(x\to\infty,\quad f(x)\to-\infty\)
(d) Maximum real zeros: \(3\)
(e) Maximum turning points: \(2\)
(f) Actual real zeros: \(x=-2,\quad x=4\)
(g) Crosses or touches:
\(x=-2\): Touches (multiplicity 2).
\(x=4\): Crosses (multiplicity 1).
\(\textbf{14)}\) \(f(x)=x^2-6x+9\)(a) Degree: \(2\)
(b) Leading coefficient: \(1\)
(c) End behavior:
\(x\to-\infty,\quad f(x)\to\infty\)
\(x\to\infty,\quad f(x)\to\infty\)
(d) Maximum real zeros: \(2\)
(e) Maximum turning points: \(1\)
(f) Actual real zeros: \(x=3\)
(g) Crosses or touches: Touches at \(x=3\) (multiplicity 2).
(b) Leading coefficient: \(1\)
(c) End behavior:
\(x\to-\infty,\quad f(x)\to\infty\)
\(x\to\infty,\quad f(x)\to\infty\)
(d) Maximum real zeros: \(2\)
(e) Maximum turning points: \(1\)
(f) Actual real zeros: \(x=3\)
(g) Crosses or touches: Touches at \(x=3\) (multiplicity 2).
\(\textbf{15)}\) \(f(x)=-2(x-1)(x+2)(x-3)\)(a) Degree: \(3\)
(b) Leading coefficient: \(-2\)
(c) End behavior:
\(x\to-\infty,\quad f(x)\to\infty\)
\(x\to\infty,\quad f(x)\to-\infty\)
(d) Maximum real zeros: \(3\)
(e) Maximum turning points: \(2\)
(f) Actual real zeros: \(x=-2,\quad x=1,\quad x=3\)
(g) Crosses or touches: Crosses at all three zeros (multiplicity 1 each).
(b) Leading coefficient: \(-2\)
(c) End behavior:
\(x\to-\infty,\quad f(x)\to\infty\)
\(x\to\infty,\quad f(x)\to-\infty\)
(d) Maximum real zeros: \(3\)
(e) Maximum turning points: \(2\)
(f) Actual real zeros: \(x=-2,\quad x=1,\quad x=3\)
(g) Crosses or touches: Crosses at all three zeros (multiplicity 1 each).
\(\textbf{16)}\) \(f(x)=(x^2+1)(x-2)\)(a) Degree: \(3\)
(b) Leading coefficient: \(1\)
(c) End behavior:
\(x\to-\infty,\quad f(x)\to-\infty\)
\(x\to\infty,\quad f(x)\to\infty\)
(d) Maximum real zeros: \(3\)
(e) Maximum turning points: \(2\)
(f) Actual real zeros: \(x=2\)
(g) Crosses or touches: Crosses at \(x=2\) (multiplicity 1).
(b) Leading coefficient: \(1\)
(c) End behavior:
\(x\to-\infty,\quad f(x)\to-\infty\)
\(x\to\infty,\quad f(x)\to\infty\)
(d) Maximum real zeros: \(3\)
(e) Maximum turning points: \(2\)
(f) Actual real zeros: \(x=2\)
(g) Crosses or touches: Crosses at \(x=2\) (multiplicity 1).
\(\textbf{17)}\) \(f(x)=(x-2)^2(x+1)^2\)(a) Degree: \(4\)
(b) Leading coefficient: \(1\)
(c) End behavior:
\(x\to-\infty,\quad f(x)\to\infty\)
\(x\to\infty,\quad f(x)\to\infty\)
(d) Maximum real zeros: \(4\)
(e) Maximum turning points: \(3\)
(f) Actual real zeros: \(x=-1,\quad x=2\)
(g) Crosses or touches: Touches at both zeros (multiplicity 2 each).
(b) Leading coefficient: \(1\)
(c) End behavior:
\(x\to-\infty,\quad f(x)\to\infty\)
\(x\to\infty,\quad f(x)\to\infty\)
(d) Maximum real zeros: \(4\)
(e) Maximum turning points: \(3\)
(f) Actual real zeros: \(x=-1,\quad x=2\)
(g) Crosses or touches: Touches at both zeros (multiplicity 2 each).
\(\textbf{18)}\) \(f(x)=-x^5+4x^3\)(a) Degree: \(5\)
(b) Leading coefficient: \(-1\)
(c) End behavior:
\(x\to-\infty,\quad f(x)\to\infty\)
\(x\to\infty,\quad f(x)\to-\infty\)
(d) Maximum real zeros: \(5\)
(e) Maximum turning points: \(4\)
(f) Actual real zeros: \(x=-2,\quad x=0,\quad x=2\)
(g) Crosses or touches:
\(x=-2\): Crosses (multiplicity 1).
\(x=0\): Crosses with flattening (multiplicity 3).
\(x=2\): Crosses (multiplicity 1).
(b) Leading coefficient: \(-1\)
(c) End behavior:
\(x\to-\infty,\quad f(x)\to\infty\)
\(x\to\infty,\quad f(x)\to-\infty\)
(d) Maximum real zeros: \(5\)
(e) Maximum turning points: \(4\)
(f) Actual real zeros: \(x=-2,\quad x=0,\quad x=2\)
(g) Crosses or touches:
\(x=-2\): Crosses (multiplicity 1).
\(x=0\): Crosses with flattening (multiplicity 3).
\(x=2\): Crosses (multiplicity 1).
\(\textbf{19)}\) \(f(x)=3x^6-12x^4\)(a) Degree: \(6\)
(b) Leading coefficient: \(3\)
(c) End behavior:
\(x\to-\infty,\quad f(x)\to\infty\)
\(x\to\infty,\quad f(x)\to\infty\)
(d) Maximum real zeros: \(6\)
(e) Maximum turning points: \(5\)
(f) Actual real zeros: \(x=-2,\quad x=0,\quad x=2\)
(g) Crosses or touches:
\(x=-2\): Crosses (multiplicity 1).
\(x=0\): Touches (multiplicity 4).
\(x=2\): Crosses (multiplicity 1).
(b) Leading coefficient: \(3\)
(c) End behavior:
\(x\to-\infty,\quad f(x)\to\infty\)
\(x\to\infty,\quad f(x)\to\infty\)
(d) Maximum real zeros: \(6\)
(e) Maximum turning points: \(5\)
(f) Actual real zeros: \(x=-2,\quad x=0,\quad x=2\)
(g) Crosses or touches:
\(x=-2\): Crosses (multiplicity 1).
\(x=0\): Touches (multiplicity 4).
\(x=2\): Crosses (multiplicity 1).
\(\textbf{20)}\) \(f(x)=(x-1)^3(x+2)^2\)(a) Degree: \(5\)
(b) Leading coefficient: \(1\)
(c) End behavior:
\(x\to-\infty,\quad f(x)\to-\infty\)
\(x\to\infty,\quad f(x)\to\infty\)
(d) Maximum real zeros: \(5\)
(e) Maximum turning points: \(4\)
(f) Actual real zeros: \(x=-2,\quad x=1\)
(g) Crosses or touches:
\(x=-2\): Touches (multiplicity 2).
\(x=1\): Crosses with flattening (multiplicity 3).
(b) Leading coefficient: \(1\)
(c) End behavior:
\(x\to-\infty,\quad f(x)\to-\infty\)
\(x\to\infty,\quad f(x)\to\infty\)
(d) Maximum real zeros: \(5\)
(e) Maximum turning points: \(4\)
(f) Actual real zeros: \(x=-2,\quad x=1\)
(g) Crosses or touches:
\(x=-2\): Touches (multiplicity 2).
\(x=1\): Crosses with flattening (multiplicity 3).
See Related Pages
\(\bullet\text{ Adding and Subtracting Polynomials}\)
\(\,\,\,\,\,\,\,\,(4d+7)-(2d-5)…\)
\(\bullet\text{ Multiplying Polynomials}\)
\(\,\,\,\,\,\,\,\,(x+2)(x^2+3x-5)…\)
\(\bullet\text{ Dividing Polynomials}\)
\(\,\,\,\,\,\,\,\,(x^3-8)\div(x-2)…\)
\(\bullet\text{ Dividing Polynomials (Synthetic Division)}\)
\(\,\,\,\,\,\,\,\,(x^3-8)\div(x-2)…\)
\(\bullet\text{ Synthetic Substitution}\)
\(\,\,\,\,\,\,\,\,f(x)=4x^4-3x^2+8x-2…\)
\(\bullet\text{ End Behavior}\)
\(\,\,\,\,\,\,\,\,\text{As }x\rightarrow\infty,\quad f(x)\rightarrow\infty…\)
\(\bullet\text{ Completing the Square}\)
\(\,\,\,\,\,\,\,\,x^2+10x-24=0…\)
\(\bullet\text{ Quadratic Formula and the Discriminant}\)
\(\,\,\,\,\,\,\,\,x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}…\)
\(\bullet\text{ Complex Numbers}\)
\(\,\,\,\,\,\,\,\,i=\sqrt{-1}…\)
\(\bullet\text{ Multiplicity of Roots}\)
\(\,\,\,\,\,\,\,\,\)
\(…\)
\(\bullet\text{ Rational Zero Theorem}\)
\(\,\,\,\,\,\,\,\,\pm1,\pm2,\pm3,\pm4,\pm6,\pm12…\)
\(\bullet\text{ Descartes Rule of Signs}\)
\(\,\)
\(\bullet\text{ Roots and Zeroes}\)
\(\,\,\,\,\,\,\,\,\text{Solve for }x.\ 3x^2+4x=0…\)
\(\bullet\text{ Linear Factored Form}\)
\(\,\,\,\,\,\,\,\,f(x)=(x+4)(x+1)(x-3)…\)
\(\bullet\text{ Polynomial Inequalities}\)
\(\,\,\,\,\,\,\,\,x^3-4x^2-4x+16\gt0…\)