![]() We can find out the two roots by factorizing the equation into a quadratic equation. Solution: Let us consider 3 roots in AP to be x - d, x, and x + d.įrom the equation, x 3 - 12x 2 + 39x - 28 = 0 we know, The formulas that indicate the relation between roots and the coefficients of a cubic polynomial are:Įxample: Solve the cubic equation, x 3 - 12x 2 + 39x - 28 = 0 given that its roots are in arithmetic progression. Let us say p,q, and r are 3 roots for the equation ax 3 + bx 2 + cx + d. When a cubic polynomial is solved graphically, we get to the accurate roots or when we solve the equation with the formula, we derive the roots. In most cases, there are 3 roots of a cubic polynomial but sometimes we do get two or only one. The solutions to a cubic equation are called the roots of the cubic equation. Thus, the roots of given cubic equation are: 5, (-3 + i √19) /, and (-3 - i √19) /2. Now, we will solve x 2 + 3x + 7 = 0 by the quadratic formula. Thus, by factor theorem, the given cubic equation can be written as: (x 3 - 2x 2 - 8x - 35) = (x - 5) (x 2 + 3x + 7). It is x 2 + 3x + 7 and the remainder is 0. Thus, the quotient is one order less than the given polynomial. Here, the zero of the linear factor is found by: x - 5 = 0 ⇒ x = 5. Let's use synthetic division to find the quotient. Step 5: Express the given polynomial as the product of its factors.Įxample: Solve cubic equation x 3 - 2x 2 - 8x - 35 = 0 if (x - 5) is a factor of the cubic polynomial x 3 - 2x 2 - 8x - 35.Step 4: If it is possible, factor the quadratic quotient further.Step 3: Using the division algorithm, write the given polynomial as the product of (x - a) and the quadratic quotient q(x).If the remainder is not zero, then it means that (x - a) is not a factor of p(x). Step 2: Once the division is completed the remainder should be 0.Step 1: Use the synthetic division method to divide the given polynomial p(x) by the given binomial (x − a).The solution to p(x) = 0 is "a" and the zero of the function p(x) is "a".įor degrees like 3 and 4, such as a cubic equation, factor theorem is used along with synthetic division and the steps are as follows:.The remainder is zero when p(x) is divided by (x – a).It is important to note that all the following statements apply for any polynomial p(x) when (x – a) is a factor of p(x). The formula of the factor theorem is p(x) = (x – a) q(x). As per the factor theorem, (x – a) can be considered as a factor of the polynomial p(x) of degree n ≥ 1, if and only if p(a) = 0. Step 7: Now group the coefficients with the variables to get the quotient.įactor theorem is a that links the factors of a polynomial and its zeros.Step 6: Separate the last term thus obtained which is the remainder.Step 5: Repeat the previous 2 steps until you reach the last term.Step 4: Add them and write the value below.Step 3: Bring the first coefficient down, multiply it with zero of linear factor and write it below the next coefficient.Step 2: Write the coefficients in the dividend's place and write the zero of the linear factor in the divisor's place.Step 1: Check whether the cubic polynomial is in the standard form.While solving a cubic polynomial we use the synthetic division method and the steps are: We can represent the division of two polynomials in the form: p(x)/q(x) = Q + R/(q(x)) ![]() Synthetic division is a method used to perform the division operation on polynomials when the divisor is a linear factor. ☛ Note: Alternatively, a cubic equation can be solved by the rational root theorem. Let us see how to solve cubic equations using these steps. Here, Step 2 can be done by using a combination of the synthetic division method and the factor theorem.
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