Exercise
Factor the polynomial equation
x^4-3x^2-4=0
Final Answer
Solution
First solution:
x^4-3x^2-4=0
To reach a quadratic equation, we define a new variable:
y=x^2
We set the new variable:
y^2-3y-4=0
We got a quadratic equation. Our equation coefficients are
a=1, b=-3, c=-4
We solve it with the quadratic formula. Putting the coefficients in the formula gives us
y_{1,2}=\frac{3\pm \sqrt{{(-3)}^2-4\cdot 1\cdot (-4)}}{2\cdot 1}=
=\frac{3\pm \sqrt{25}}{2}=
=\frac{3\pm 5}{2}
Hence, we get the solutions:
y_1=\frac{3+ 5}{2}=\frac{8}{2}=4
y_2=\frac{3- 5}{2}=\frac{-2}{2}=-1
Thus, the factorizing of the quadratic equation is
y^2-3y-4=
=(y-4)(y+1)
Going back to the original variable we get
=(x^2-4)(x^2+1)
We break down the first factor using Short Multiplication Formulas (third formula) and get the factorizing
=(x-2)(x+2)(x^2+1)
Second solution:
x^4-3x^2-4=
We express the second expression as sum and we get
=x^4-4x^2+x^2-4=
We extract a common factor from the first two expressions:
=x^2(x^2-4)+(x^2-4)=
Once again we extract a common factor – this time the expression in parentheses:
=(x^2-4)(x^2+1)=
We break down the first factor using Short Multiplication Formulas (third formula) and get the factorizing:
=(x-2)(x+2)(x^2+1)
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