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A trinomial is a kind of algebraic equation in terms of variables and constants. Trinomials can exist in different formats. Factoring trinomial of the form x⁴+bx²+c is similar to factoring the quadratic equation of the format ax²+bx+c. Factoring is one of the simplest ways to solve an algebraic equation. A trinomial equation is a type of polynomial that has three components. Among those three terms, no term is valued as zero. These terms are separated by basic mathematical operations like addition, subtraction, multiplication, and division.
Trinomial equation
A trinomial equation is a type of polynomial equation.
An equation having three terms in the equation is called a trinomial.
When the highest power in a trinomial equation is ‘4’, it is termed a quartic equation.
When the highest power of an equation is 4, the factors of the equation are also four different answers.
Factoring
Factorising is one of the easiest methods to solve a trinomial equation.
It is the method of finding the factors.
The breaking down of a larger number into its multiples gives the factors of the number.
For example, the multiples of 21 are 7 and 3. Thus, 7 and 3 are the factors of 21.
Factoring trinomials of the form x⁴+bx²+c
Factoring is generally used to solve any trinomial equation, whether quartic or quadratic. Any given trinomial can be solved using the following simple steps.
Consider a trinomial equation x⁴+6x²-27 of the form x⁴+bx²+c. The process is so similar to solving a quadratic trinomial equation.
Split the middle term
Split the middle term(6x²) into two factors in such a way that
The sum of the two factors equals the coefficient(6) of x² and
Their product equals the product of the x⁴ coefficient(1) and the constant(-27).
x⁴ + 6x² – 27 = 0
x⁴ + 9x² – 3x² – 27 = 0
Here, the middle term(6x²) is split into 9x² and -3x².
+9 * -3 = +1 * -27
+9 + (-3) = 6
Take the common factor
Now, note the common factor in the equation and take it out as
Grouping the first two terms(x⁴ and 9x²) separately and
The last two terms(3x² and 27) are separate.
x⁴ + 9x² – 3x² – 27 = 0
x²(x² + 9) – 3(x² + 9) = 0
Take the common factor again
Notice that the terms inside the parentheses are the same.
Take them as common factors.
x²(x² + 9) – 3(x² + 9) = 0
(x² + 9) (x² – 3) = 0
Equate to zero
Equate the two sets of parentheses to zero, separately, to find the values of ‘x’.
(x² + 9) (x² – 3) = 0
Equating (x² + 9),
x² + 9 = 0
x² = – 9
x = √-9
Equating (x² – 3),
x² – 3 = 0
x² = +3
x = √3
Root value
The factors of the given trinomial equations are +√9,-√9, +√3, -√3.
Since the highest power of the given trinomial equation is ‘4’, there are four root values as the factors of the equation.
Solved examples of factoring trinomials of the form x⁴+bx²+c
Some solved examples of factoring trinomials of the form x⁴+bx²+c are given below:
1) x⁴+18x²+80 = 0
x⁴+10x²+8x²+80 = 0
Taking common factors out,
x²(x²+10) + 8(x²+10) = 0
Taking common factors out,
(x²+10) (x²+8) = 0
Equating to zero,
x²+10 = 0
x² = -10
x = -√10, +√10
x²+8 = 0
x² = -8
x = -√8, +√8
2) x⁴-13x²+42 = 0
x⁴-6x²-7x²+42 = 0
Taking common factors out,
x²(x²-6) – 7(x²-6) = 0
Taking common factors out,
(x²-6) (x²-7) = 0
Equating to zero,
x²-6 = 0
x² = +6
x = -√6, +√6
x²-7 = 0
x² = +7
x = -√7, +√7
3) x⁴-17x²+16 = 0
x⁴-1x²-16x²+16 = 0
Taking common factors out,
x²(x²-1) – 16(x²-1) = 0
Taking common factors out,
(x²-16) (x²-1) = 0
Equating to zero,
x²-16 = 0
x² = +16
x = -4, +4
x²-1 = 0
x² = +1
x = -√1, +√1
4) 2x⁴-7x²-4 = 0
2x⁴+1x²-8x²-4 = 0
Taking common factors out,
x²(2x²+1) – 4(2x²+1) = 0
Taking common factors out,
(2x²+1) (x²-4) = 0
Equating to zero,
2x²+1 = 0
x² = -1/2
x = -√1/2, +√1/2
x²-4 = 0
x² = +4
x = -2, +2
5) 6x⁴-13x²+5 = 0
6x⁴-3x²-10x²+5 = 0
Taking common factors out,
3x²(2x²-1) – 5(2x²-1) = 0
Taking common factors out,
(2x²-1) (3x²-5) = 0
Equating to zero,
2x²-1 = 0
x² = 1/2
x = -√1/2, +√1/2
3x²-5 = 0
x² = 5/3
x = -√5/3, +√5/3
6) 6x⁴+11x²+3 = 0
6x⁴+2x²+9x²+3 = 0
Taking common factors out,
2x²(3x²+1) + 3(3x²+1) = 0
Taking common factors out,
(3x²+1) (2x²+3) = 0
Equating to zero,
3x²+1 = 0
x² = -1/3
x = -√1/3, +√1/3
2x²+3 = 0
x² = -3/2
x = -√3/2, +√3/2
Conclusion
An algebraic equation having three different components is called a trinomial equation. It has various variables and constants. The value of variables or their coefficients is a non-zero number. Factoring trinomials of the form x⁴+bx²+c is one of the methods to find the root factors of the equation. It gives the value of the variable ‘x’ in the equation.
