Practice Factoring General Polynomials

Factor completely.

1. \(y^{10}+7y^5-8=\)

2. \(-5h^4-15h^3-10h^2=\)

3. \(x^2-3xy-10y^2=\)

4. \(n^8-5n^4-6=\)

Source: Khan Academy, https://www.khanacademy.org/math/algebra-home/alg-polynomials/alg-factoring-polynomials-quadratic-forms/e/factoring_polynomials_2
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1. We will factor ‍\(y^{10}{+7}y^5{-8}\) as \((y^5+a)(y^5+b)\).

   We need to find numbers \(a\) and ‍\(b\) such that ‍\({a+b}={7}\) and \({ab} ={-8}\).

   \(\begin{aligned}
   (+8)+(-1)=7
   \\
   (+8)(-1)=-8
   \end{aligned}\)
   

   So the integers we are looking for are +8 and -1.

   In conclusion, \(y^{10}+7y^5-8=(y^5+8)(y^5-1)\).

2. Take a common factor of \(-5h^2\)..

   \(-5h^4-15h^3-10h^2=-5h^2(h^2+3h+2)\)

   Now let's factor \(V\).

   \(h^2+3h+2=(h+1)(h+2)\)

   In conclusion, \(\begin{aligned}
   -5h^4-15h^3-10h^2 =-5h^2(h^2+3h+2)
   \\\\
   &=-5h^2(h+1)(h+2)
   \end{aligned}\).

3. We will factor ‍\(x^2{-3}xy{-10}y^2\) as \((x+ay)(x+by)\).

   We need to find numbers \(a\) and ‍\(b\) such that ‍\({a+b}={-3}\) and \({ab} ={-10}\).

   \(\begin{aligned}
   (+2)+(-5)=-3
   \\
   (+2)(-5)=-10
   \end{aligned}\)
   

   So the integers we are looking for are +2 and -5.

   In conclusion, \(x^2-3xy-10y^2=(x+2y)(x-5y)\).

4. We will factor ‍\(n^8{-5}n^4{-6}\) as \((n^4+a)(n^4+b)\).

   We need to find numbers \(a\) and ‍\(b\) such that ‍\({a+b}={-5}\) and \({ab} ={-6}\).

   \(\begin{aligned}
   &(+1)+(-6)=-5
   \\\\
   &(+1)(-6)=-6
   \end{aligned}\)

   So the integers we are looking for are +1 and -6.

   In conclusion, \(n^8-5n^4-6=(n^4+1)(n^4-6)\).