Use synthetic division to compute \(\left (3x^2-2x+1 \right ) \div \left (x-1\right )\). Identify the quotient and remainder. Write the divisor, quotient and remainder in the form \(p(x) = d(x) \, q(x) + r(x),\,\), as in Theorem polydivthm.

\(3x^2-2x+1 = \left (\answer {x-1}\right )\left (\answer {3x+1}\right )+\answer {2}\)

Use synthetic division to compute \(\left (x^2-5 \right ) \div \left (x-5\right )\). Identify the quotient and remainder. Write the divisor, quotient and remainder in the form \(p(x) = d(x) \, q(x) + r(x),\,\), as in Theorem polydivthm.

\(x^2-5 = \left (\answer {x-5}\right )\left (\answer {x+5}\right ) + \answer {20}\)

Use synthetic division to compute \(\left (3-4t-2t^2 \right ) \div \left (t+1\right )\). Identify the quotient and remainder. Write the divisor, quotient and remainder in the form \(p(t) = d(t) \, q(t) + r(t),\,\), as in Theorem polydivthm.

\(3-4t-2t^2 = \left (\answer {t+1}\right )\left (\answer {-2t-2}\right )+\answer {5}\)

Use synthetic division to compute \(\left (4t^2-5t +3\right ) \div \left (t+3\right )\). Identify the quotient and remainder. Write the divisor, quotient and remainder in the form \(p(t) = d(t) \, q(t) + r(t),\,\), as in Theorem polydivthm.

\(4t^2-5t +3 = \left (\answer {t+3}\right )\left (\answer {4t-17}\right ) + \answer {54}\)

Use synthetic division to compute \(\left (z^3 + 8 \right ) \div \left (z+2\right )\). Identify the quotient and remainder. Write the divisor, quotient and remainder in the form \(p(z) = d(z) \, q(z) + r(z),\,\), as in Theorem polydivthm.

\(z^3 + 8 = \left (\answer {z+2}\right )\left (\answer {z^2-2z+4}\right ) + \answer {0}\)

Use synthetic division to compute \(\left (4z^3 +2z-3 \right ) \div \left (z -3\right )\). Identify the quotient and remainder. Write the divisor, quotient and remainder in the form \(p(z) = d(z) \, q(z) + r(z),\,\), as in Theorem polydivthm.

\(4z^3 +2z-3 = \left (\answer {z-3}\right )\left (\answer {4z^2+12z+38}\right ) + \answer {111}\)

Use synthetic division to compute \(\left (18x^2-15x-25\right ) \div \left (x - \frac {5}{3} \right )\). Identify the quotient and remainder. Write the divisor, quotient and remainder in the form \(p(x) = d(x) \, q(x) + r(x),\,\), as in Theorem polydivthm.

\(18x^2-15x-25 = \left (\answer {x - \frac {5}{3}}\right )\left (\answer {18x+15}\right )+\answer {0}\)

Use synthetic division to compute \(\left (4x^2-1 \right ) \div \left (x - \frac {1}{2} \right )\). Identify the quotient and remainder. Write the divisor, quotient and remainder in the form \(p(x) = d(x) \, q(x) + r(x),\,\), as in Theorem polydivthm.

\(4x^2-1 = \left (\answer {x - \frac {1}{2}}\right )\left (\answer {4x+2}\right ) + \answer {0}\)

Use synthetic division to compute \(\left (2t^3+t^2+2t+1 \right ) \div \left (t + \frac {1}{2} \right )\). Identify the quotient and remainder. Write the divisor, quotient and remainder in the form \(p(t) = d(t) \, q(t) + r(t),\,\), as in Theorem polydivthm.

\(2t^3+t^2+2t+1 = \left (\answer {t + \frac {1}{2}} \right )\left (\answer {2t^2+2}\right )+\answer {0}\)

Use synthetic division to compute \(\left (3t^3 - t + 4 \right ) \div \left (t - \frac {2}{3} \right )\). Identify the quotient and remainder. Write the divisor, quotient and remainder in the form \(p(t) = d(t) \, q(t) + r(t),\,\), as in Theorem polydivthm.

\(3t^3 - t + 4 = \left (\answer {t - \frac {2}{3}}\right )\left (\answer {3t^2+2t+\frac {1}{3}}\right ) + \answer {\frac {38}{9}}\)

Use synthetic division to compute \(\left (2z^3 - 3z +1 \right ) \div \left (z - \frac {1}{2} \right )\). Identify the quotient and remainder. Write the divisor, quotient and remainder in the form \(p(z) = d(z) \, q(z) + r(z),\,\), as in Theorem polydivthm.

\(2z^3 - 3z +1 = \left (\answer {z - \frac {1}{2}} \right ) \left (\answer {2z^2+z-\frac {5}{2}}\right ) + \answer {-\frac {1}{4}}\)

Use synthetic division to compute \(\left (4z^4-12z^3+13z^2 -12z+9\right ) \div \left (z - \frac {3}{2} \right )\). Identify the quotient and remainder. Write the divisor, quotient and remainder in the form \(p(z) = d(z) \, q(z) + r(z),\,\), as in Theorem polydivthm.

\(4z^4-12z^3+13z^2 -12z+9 = \left (\answer {z - \frac {3}{2}}\right )\left (\answer {4z^3-6z^2+4z-6}\right ) + \answer {0}\)

Use synthetic division to compute \(\left (x^4-6x^2+9 \right ) \div \left (x -\sqrt {3} \right )\). Identify the quotient and remainder. Write the divisor, quotient and remainder in the form \(p(x) = d(x) \, q(x) + r(x),\,\), as in Theorem polydivthm.

\(x^4-6x^2+9 = \left (\answer {x -\sqrt {3}} \right ) \left (\answer {x^3+\sqrt {3} \,x^2-3x-3\sqrt {3}}\right ) + \answer {0}\)

Use synthetic division to compute \(\left (x^6-6x^4+12x^2-8\right ) \div \left (x +\sqrt {2} \right )\). Identify the quotient and remainder. Write the divisor, quotient and remainder in the form \(p(x) = d(x) \, q(x) + r(x),\,\), as in Theorem polydivthm.

\(\left (x^6-6x^4+12x^2-8\right ) = \left (x +\sqrt {2} \right ) \left (x^5-\sqrt {2} \, x^4-4x^3+4\sqrt {2} \, x^2+4x-4\sqrt {2}\right ) + 0\)
Find \(p(c)\) using the Remainder Theorem. If \(p(c) = 0\), use the Factor Theorem to partially factor the polynomial function.

\(p(x) = 2x^2 - x + 1\), \(c = 4\)

\(p(4) = \answer {29}\)

Find \(p(c)\) using the Remainder Theorem. If \(p(c) = 0\), use the Factor Theorem to partially factor the polynomial function.

\(p(x) = 4x^2-33x-180\), \(c = 12\)

\(p(12) =\answer {0}\), \(p(x) = \answer {(x-12)(4x+15)}\)

Find \(p(c)\) using the Remainder Theorem. If \(p(c) = 0\), use the Factor Theorem to partially factor the polynomial function.

\(p(t) = 2t^3 - t + 6\), \(c=-3\)

\(p(-3)= \answer {-45}\)

Find \(p(c)\) using the Remainder Theorem. If \(p(c) = 0\), use the Factor Theorem to partially factor the polynomial function.

\(p(t) = t^3+2t^2+3t+4\), \(c =-1\)

\(p(-1)=\answer {2}\)

Find \(p(c)\) using the Remainder Theorem. If \(p(c) = 0\), use the Factor Theorem to partially factor the polynomial function.

\(p(z) =3z^3-6z^2+4z-8\), \(c=2\)

\(p(2) = \answer {0}\), \(p(z)= \answer {(z-2) \left (3z^2+4\right )}\)

Find \(p(c)\) using the Remainder Theorem. If \(p(c) = 0\), use the Factor Theorem to partially factor the polynomial function.

\(p(z) = 8z^3+12z^2+6z+1\), \(c =-\frac {1}{2}\)

\(p\left (-\frac {1}{2}\right ) = 0\), \(p(z) = \left (z+\frac {1}{2}\right )\left (8z^2+8z+2\right )\)
Find \(p(c)\) using the Remainder Theorem. If \(p(c) = 0\), use the Factor Theorem to partially factor the polynomial function.

\(p(x) = x^4 - 2x^2+4\), \(c=\frac {3}{2}\)

\(p\left (\frac {3}{2}\right ) = \frac {73}{16}\)
Find \(p(c)\) using the Remainder Theorem. If \(p(c) = 0\), use the Factor Theorem to partially factor the polynomial function.

\(p(x) = 6x^4-x^2+2\), \(c =-\frac {2}{3}\)

\(p\left (-\frac {2}{3}\right ) = \frac {74}{27}\)
Find \(p(c)\) using the Remainder Theorem. If \(p(c) = 0\), use the Factor Theorem to partially factor the polynomial function.

\(p(t) = t^4 +t^3-6t^2-7t-7\), \(c=-\sqrt {7}\)

\(p(-\sqrt {7}) = 0\), \(p(t) = (t+\sqrt {7})\left (t^3+(1-\sqrt {7}) t^2+(1-\sqrt {7})t-\sqrt {7} \right )\)
Find \(p(c)\) using the Remainder Theorem. If \(p(c) = 0\), use the Factor Theorem to partially factor the polynomial function.

\(p(t) = t^2-4t+1\), \(c =2-\sqrt {3}\)

\(p(2-\sqrt {3}) =0\), \(p(t) = (t-(2-\sqrt {3}))(t-(2+\sqrt {3})) \)
In this exercise you are given a polynomial function and one of its zeros. Find the remaining real zeros and factor the polynomial.

\(x^{3} - 6x^{2} + 11x - 6, \;\; c = 1\)

\(x^{3} - 6x^{2} + 11x - 6 = (x - 1)(x - 2)(x - 3)\)
In this exercise you are given a polynomial function and one of its zeros. Find the remaining real zeros and factor the polynomial.

\(x^{3} - 24x^{2} + 192x - 512, \;\; c = 8\)

When we factor the polynomial we get \(x^{3} - 24x^{2} + 192x - 512 = \answer {(x - 8)^{3}}\)

In this exercise you are given a polynomial function and one of its zeros. Find the remaining real zeros and factor the polynomial.

\(3t^{3} + 4t^{2} - t - 2, \;\; c = \frac {2}{3}\)

\(3t^{3} + 4t^{2} - t - 2 = 3\left (t - \frac {2}{3}\right )(t + 1)^{2}\)
In this exercise you are given a polynomial function and one of its zeros. Find the remaining real zeros and factor the polynomial.

\(2t^3-3t^2-11t+6, \;\; c=\frac {1}{2}\)

\(2t^3-3t^2-11t+6 = 2\left (t-\frac {1}{2}\right )(t+2)(t-3)\)
In this exercise you are given a polynomial function and one of its zeros. Find the remaining real zeros and factor the polynomial.

\(z^3+2z^2-3z-6, \;\; c = -2\)

\(z^3+2z^2-3z-6 = (z+2)(z+\sqrt {3})(z-\sqrt {3})\)
In this exercise you are given a polynomial function and one of its zeros. Find the remaining real zeros and factor the polynomial.

\(2z^3-z^2-10z+5, \;\; c=\frac {1}{2}\)

\(2z^3-z^2-10z+5=2\left (z-\frac {1}{2}\right )(z+\sqrt {5})(z-\sqrt {5})\)
In this exercise you are given a polynomial function and one of its zeros. Find the remaining real zeros and factor the polynomial.

\(4x^{4} - 28x^{3} + 61x^{2} - 42x + 9\), \(c = \frac {1}{2}\) is a zero of multiplicity 2

\(4x^{4} - 28x^{3} + 61x^{2} - 42x + 9 = 4\left (x - \frac {1}{2} \right )^{2}(x - 3)^{2}\)
In this exercise you are given a polynomial function and one of its zeros. Find the remaining real zeros and factor the polynomial.

\(t^5+2t^4-12t^3-38t^2-37t-12\), \(c=-1\) is a zero of multiplicity 3

When we factor the polynomial we get \(t^5+2t^4-12t^3-38t^2-37t-12 = \answer {(t+1)^3(t+3)(t-4)}\)

In this exercise you are given a polynomial function and one of its zeros. Find the remaining real zeros and factor the polynomial.

\(125z^{5} - 275z^{4} - 2265z^{3} - 3213z^{2} - 1728z - 324\), \(c = -\frac {3}{5}\) is a zero of multiplicity 3

\(125z^{5} - 275z^{4} - 2265z^{3} - 3213z^{2} - 1728z - 324\) = \(125\left (z + \frac {3}{5} \right )^{3}(z + 2)(z - 6)\)
In this exercise you are given a polynomial function and one of its zeros. Find the remaining real zeros and factor the polynomial.

\(x^{2} - 2x - 2, \;\; c = 1 - \sqrt {3}\)

\(x^{2} - 2x - 2 = (x - (1 - \sqrt {3}))(x - (1 + \sqrt {3}))\)
Find a quadratic polynomial with integer coefficients which has \(x = \frac {3}{5} \pm \frac {\sqrt {29}}{5}\) as its real zeros.

\(p(x) = 5x^{2} - 6x - 4\)

For \(f(x) = x^3 + 4x^2-5x-14\), show \(f(-3-\sqrt {2}) = 0\) and \(f(-3+\sqrt {2}) = 0\) two ways:

  1. By direct substitution.
  2. Using synthetic division and the Factor Theorem
Let \(f(x) = a_{n} x^{n} + a_{n-1} x^{n-1} + \ldots + a_{2} x^{2} + a_{1} x + a_{0}\) be a polynomial function with the property that \( a_{n}+a_{n-1} + \ldots + a_{1} + a_{0} = 0\). (That is, the sum of the coefficients and the constant term is \(0\).)

Prove that \((x-1)\) is a factor of \(f(x)\).

Show \(f(1) = 0\) and invoke the Factor Theorem.
Verify the result in number oneisazeroex with the functions: \(f(x) = x^3 - 2x + 1\) and \(f(x) = 3x^4-x-2\).

  • For \(f(x) = x^3 - 2x + 1\), the coefficients \(1+(-2) + 1 = 0\) and \(f(x) = (x-1)(x^2+x-1)\).
  • For \(f(x) = 3x^4-x-2\) the coefficients \(3+(-1)+(-2) = 0\) and \(f(x) = (x-1)(3x^3+3x^2+3x+2)\).
Suppose \(a\) is a nonzero real number. Find the quotients below, using synthetic division as required.
  • \(\frac {x - a}{x-a} = \answer {1}\)
  • \(\frac {x^2 - a^2}{x-a} = \answer {x + a}\)
  • \(\frac {x^3 - a^3}{x-a} = \answer { x^2+ax+a^2}\)
  • \(\frac {x^4 - a^4}{x-a} = \answer { x^3 + ax^2 + a^2x + a^3}\)
  • \(\frac {x^5 - a^5}{x-a} = \answer {x^4+ax^3 + a^2x^2+a^3x+a^4}\)

Based on the pattern that evolves, find the quotient: \(\frac {x^{10} - a^{10}}{x-a}\). What about \(\frac {x^{n} - a^{n}}{x-a}\)?

Following the pattern:
  • \(\frac {x^{10} - a^{10}}{x-a} = x^{9} + ax^8 + a^2x^7+a^3x^6+a^4x^5+a^5x^4+a^6x^3+a^7x^2+a^8x+a^9\)
  • \(\frac {x^{n} - a^{n}}{x-a} = x^{n-1} + ax^{n-2} + a^2x^{n-3} + \ldots + a^{n-2} x + a^{n-1}\)
Use your result from number monomialdiffquotex to rewrite the sum: \(1 + r + r^2 + \dots + r^{n-2} + r^{n-1}\) as a quotient. What assumptions need to be made about \(r\)?

Put \(x=1\) and \(a = r\) so that \(1 + r + r^2 + \dots + r^{n-2} + r^{n-1} = \frac {1 - r^{n}}{1-r}\). Here, \(r \neq 1\) as otherwise we’d be dividing by \(0\).