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In this lesson, we will learn how to solve cubic equations using the Remainder Theorem and the Factor Theorem.
What is the Remainder Theorem?
If a polynomial, f(x), is divided by x - k, the remainder is equal to f(k).
What is the Factor Theorem?
x - k is a factor of the polynomial f(x) if and only if f(k) = 0
How to solve cubic equations using the Factor Theorem?
In these lessons, we will consider how to solve cubic equations of the form
= 0 where p, q, r
are constants by using the Factor Theorem and Synthetic Division.
Find the roots of f(x) = 2x3 + 3x2 – 11x – 6 = 0, given that it has at least one integer root.
Since the constant in the given equation is a 6, we know that the integer root must be a factor of 6. The possible values are
Step 1: Use the factor theorem to test the possible values by trial and error.
f(1) = 2 + 3 – 11 – 6 ≠ 0
f(–1) = –2 + 3 + 11 – 6 ≠ 0
f(2) = 16 + 12 – 22 – 6 = 0
We find that the integer root is 2.
Step 2: Find the other roots either by inspection or by synthetic division
2x3 + 3x2 – 11x – 6
= (x – 2)(ax2 + bx + c)
= (x – 2)(2x2 + bx + 3)
= (x – 2)(2x2 + 7x + 3)
= (x – 2)(2x + 1)(x +3)
So, the roots are
Solve the cubic equation x3 – 7x2 + 4x + 12 = 0
Let f(x) = x3 – 7x2 + 4x + 12
The possible values are
We find that f(–1) = –1 – 7 – 4 + 12 = 0
So, (x + 1) is a factor of f(x)
x3 – 7x2 + 4x + 12
= (x + 1)(x2 – 8x + 12)
= (x + 1)(x – 2)(x – 6)
So, the roots are –1, 2, 6
How to use the Factor Theorem to factor polynomials?
1) Factor P(x) =
− 10x + 8
2) Factor P(x) =
+ x + 12
What are The Remainder Theorem and the Factor Theorem?
How to use the Theorems to find the linear factorization of a polynomial?
Example: Factor F(x) =
− 5x + 6
How to use the Factor Theorem to solve a cubic equation?
If f(x) is a polynomial and f(p) = 0 then x - p is a factor of f(x)
Example: Solve the equation 2x3
− 10 = 23x
How to solve Cubic Equations using the Factor theorem and Long Division?
Example: Find the roots of the cubic equation 2x3
+ 7x − 1 = 0
How to solve Cubic Equations using the Factor theorem and Synthetic Division?
Example: Show that x + 3 is a factor of x3
− 19x − 30 = 0. Then find the remaining factors of f(x)
How to solve cubic equation problems?
− 17x = x3
Step 1: Set one side of equation equal to 0.
Step 2: Collect like terms.
Step 3: Factorize using the Factor Theorem and Long Division
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